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Skeptical answer

One could give a skeptic's answer for any form of this question in the context of a political campaign: presumably the candidates feel this answer will be more likely to get them nominated by their party in their campaign for President of the United States.

However, although this position is not made explicit, it is consistent with the idea included in the 2016 Democratic Party platform that describes healthcare as a right.

Stated reason

Since you mention Joe Biden in particular, he has said in a recent interview:

“I think undocumented people need to have a means by which they can be covered when they’re sick,” he said in a CNN interview, adding, “This is just common decency.”

“In an emergency they should have health care. Everybody should,” he added. "How do you say 'You're undocumented, I'm gonna let you die, man?'"

text via The Hill, original interview at CNN.

Democratic party platform

The Democratic party platform from 2016 states (emphasis mine):

Democrats believe that health care is a right, not a privilege, and our health care system should put people before profits.

This could possibly be interpreted as health care being a right for American citizens, but it could also be interpreted as health care being a human right, and if illegal immigrants are humans then they would be deserving of that right. The latter interpretation is also consistent with the Democratic Party's position on immigration, which treats the current system as broken and acknowledges the role that undocumented immigrants play in the US economy:

Those immigrants already living in the United States, who are assets to their communities and contribute so much to our country, should be incorporated completely into our society through legal processes that give meaning to our national motto: E Pluribus Unum.

and specifically on healthcare:

We will work to ensure that all Americans—regardless of immigration status—have access to quality health care. That means expanding community health centers, allowing all families to buy into the Affordable Care Act exchanges, supporting states that open up their public health insurance programs to all persons, and finally enacting comprehensive immigration reform.

I would add that this platform position does not fit the promise of "free" health care in the title of this question, but that promise is not necessarily implied by the question asked of the candidates, either.

Status quo

The New York Times has a recent article discussing the status quo for healthcare for illegal immigrants. Currently federal funds are not supposed to be spent for healthcare for undocumented immigrants, and they are ineligible at the federal level for Medicare and Medicaid, although there are some states that do provide coverage.

However, there currently is indirect federal funding for health care for undocumented immigrants through centers that treat everyone regardless of circumstance and indirectly through hospitals that are ethically and morally required to treat ill people, whose cost of care is absorbed by those who are insured. Essentially, people in the United States illegally who do not have insurance are in a similar situation to citizens who do not have insurance: the care they do have access to is emergency care rather than preventative care.

Biden's position relative to the status quo is not entirely clear, because if he limits his stance to emergency care, then emergency care already exists. I am unaware of and did not find specific proposals from any of the candidates for how they would change the status quo.

Summary

It seems like the field of Democrats running for President is moving towards treating healthcare as a human right. The candidates may also want to further separate themselves from US government policies that currently house immigrants, and particularly children, in poor conditions. At the moment, there are not specific policies proposed by the candidates to change the status quo, and their position is consistent with the Democratic Party platform from the last presidential election cycle.

From the transcript of the second night of the debate, we can see that both Biden and Buttigieg addressed this question. I'm not aware of any public statements from other candidates on this issue, but my suspicion is that they would likely agree with these points:

BUTTIGIEG: Because our country is healthier when everybody is healthier. And remember, we’re talking about something people are getting a–given a chance to buy into. In the same way that there are undocumented immigrants in my community who pay, they pay sales taxes, they pay property taxes directly or indirectly. This is not about a handout. This is an insurance program. And we do ourselves no favor by having 11 million undocumented people in our country be unable to access healthcare

Biden states similar points and adds:

BIDEN: Yes. You cannot let–as the mayor said, you cannot let people who are sick, no matter where they come from, no matter what their status, go uncovered. You can’t do that. It’s just going to be taking care of, period. You have to. It’s the humane thing to do

To summarize, there are 3 main points:

1. It is better for our country if the people here are healthy. Denying undocumented immigrants basic medical care could lead to outbreaks of disease and a drain on the economy as their (often US Citizen) children are forced to care for them. Furthermore, prompt treatment and preventative care reduce overall healthcare costs.


2. With the government healthcare plans under discussion, undocumented immigrants would pay into the system through their taxes and so there's no contradiction in letting them receive the benefits they've paid for.


3. Finally, it's the humane thing to do. If someone is sick or dying, we shouldn't demand to see their papers or ask for cash up front.

First off, the word "free" that was originally in the question is a complete misnomer at best (at worst it would be putting words in their mouths). Nothing is free. Most universal coverage schemes are paid for at least partially through some kind of income or payroll tax, which every employee/employer pays (even if the workers in question are not citizens or are using fake SSN's). So undocumented immigrants for the most part would be paying into these systems, and thus would not be getting anything for "free" any more than anyone else would.

That being said, there are three basic arguments for true Universal coverage. They are generally used together, so look at this as a complete philosophical package, not a menu. You may not necessarily agree with it all, but this is the logic I see used:

1. 
   Its the morally right thing to do. Jesus himself commanded his followers to treat the sick multiple times. He seemed quite insistent on it. See Matthew 25 ("I was sick and you took care of me"), and Matthew 10:8:

Cure the sick, raise the dead, cleanse the lepers, cast out demons. You received without payment; give without payment.

Some Christians may think they have reasons to dismiss the relevance of these directions, but this is a big deal to Christian liberals like Pete Buttigieg, Joe Biden, and Corey Booker.

Of course these values weren't pulled out of thin air. Nearly all other religions, philosophies, and ethics codes, have some version of the Golden Rule. The basic principles of humanism many atheists are philosophically aligned with demand taking care of the sick and injured. There are certainly some moral systems out there where's it just fine to deny people medical treatment because of who they are, but they are very few. This one is about as universal a human moral code as they come.

2. 
   It is the safe thing to do for yourself and those you do care about.

You may think you and your family are a different class of person than immigrants, but diseases don't. If there are legions of untreated sick people running around the USA, that's a danger for all of us.

3. Its the financially responsible thing to do.

Hospitals are required by law to treat anyone who shows up (see #1 above if you don't like that). Those who can't pay are essentially paid for by higher costs paid by those of us who have coverage. Hospitals are also the most expensive place in our entire medical system. This means the financially stupidest decision possible is to use hospitals as the only place where US residents have universal access to treatment (regardless of status or ability to pay), and that's exactly what we are doing right now.

It would save all of us collectively a lot of money^* if everyone just had coverage down at the general practitioner level, which is the cheapest part of our system.

^{* - Theoretically. However, the relative expense of our system against those of countries with universal coverage makes a pretty strong case.}

There is another answer which should be more convincing to the right wing.

It's cheaper.

If you need to track who's eligible for treatment, deal with co-pays, and all the other paperwork associated with US healthcare, the administration involved is significant. So significant in fact that 30% of the cost of treatment is the cost of administration. The UK and Canada, both of whom have free-at-the-point-of-use healthcare, spend around 15% on administration, and both see that as something which they would like to reduce.

Of course we don't know exactly how many illegal immigrants there are in the US, but the figure as of 2014 was estimated at 14 million, out of a US population of 318 million. If we (naively) assume that immigrants have roughly the same healthcare needs as US citizens, that would require a 4.4% increase in healthcare funding. However the cost saving from eliminating administration gives a potential benefit of 15%.

So the paradoxical net result is that you get 10% extra money for healthcare by treating more people, because the money you save on managers, laywers and accountants can be spent directly on healthcare.

Because they think it's the right thing to do. Just like they feel it's wrong to deny water to someone who is thirsty, or deny food to someone going hungry, they feel it's wrong to deny care to someone who is injured or sick.

There are probably parallel philosophies in play here. I'm not making this argument myself, per se, as much as laying out what I believe may be rationales behind this -

1. The belief that access to health care should be a right - if one
   believes that access to healthcare falls under the Declaration's
   broad but not comprehensive list of inalienable rights - life,
   liberty and pursuit of happiness, then citizenship does not enter
   into the equation. All people are "endowed by their Creator,"
   according to that document, and, as such, they would have a right to
   access to health care.




2. The belief that the Constitution or our laws allow or require it -
   Our Constitution and our laws do not just apply to citizens. If
   someone with undocumented or illegal status breaks our laws,
   including murder, theft, or even our proper and legal immigration
   and visa rules, they are subject to enforcement and punitive action
   as described in our laws. With the exception of diplomatic
   immunity, it would be specious to say "I'm from country X, so I
   don't have to follow your laws. I can kill, rape and pillage with
   impunity." That is because our laws apply to everyone in our
   jurisdiction (I realize I wandered on a bit of a tangent, this was
   the main point). If our laws and policies apply to everyone under
   our jurisdiction, then a system of baseline mandated health care
   coverage would apply to anyone under our jurisdiction, regardless of
   their citizenship status. If I'm in Sweden, and chip a tooth, then
   I'm allowed access to their government-provided healthcare system.
   A Swede with a dental issue in the USA, currently, would use our
   services, and their government system would get billed at USA rates
   under our system as their citizen's "insurer." Within our borders, our rules and systems apply.




3. A basic recognition of humanity, regardless of status - If one
   believes that no one should be allowed to starve or die simply for
   lack of means or access, then it would make sense to extend what one
   might argue are basic necessities, no matter their citizenship.



4. Public health/policy - If someone illegally enters our nation or stays too long, is in
   contact with a lot of people, citizens or not, and carries an
   infectious agent - TB, Ebola, MRSA, bacterial meningitis, influenza,
   smallpox, etc. - to deny them even advanced healthcare puts the
   citizens at risk, not just non-citizens. Viruses and bacteria lack
   an understanding of national borders and do not care about
   citizenship status. They will spread and infect indiscriminately at any opportunity. A government would not only risk the life and
   welfare of their citizens, but would waste a lot more money dealing
   with outbreaks carried by a population that, by definition, is
   skilled at flying under the official "radar" of society, vs. giving
   healthcare and preventing or dealing with infectious agents at a
   more manageable stage.

Part of the point of Healthcare-as-a-right (beyond the moral issues) is that it's significantly cheaper to prevent disease/disability than it is to treat it. This requires healthcare people to take a holistic, community-based approach, not just treating individual patients who complain of X issue. Otherwise, the current system of care for the poorest US residents (get sick enough that you go to the hospital ER, and the hospital ER is required to treat you whether or not you're able to pay) is costing hospitals $38.3 billion a year (2016).

Fair or not, we're already paying for illegal immigrants' healthcare via increased hospital costs to offset these 'uncompensated care' cases. Healthcare-as-a-right just means we'd be paying less.

Preventive care improving quality of life and save 1.2 cents per 1 cent spent: 2010, UK

It depends what types of prevention: 2010, Australia

It works for American veterans to give them the PTSD support they need before entering the workforce or showing symptoms - 2018, USA

It's a lot easier to convince someone of the value of prevention as a thought experiment than in concrete situations - 2013, Netherlands

It’s a key issue that divides the Republicans from the Democrats which is why it was asked. That all the democratic candidates raised their hand to this shows the strength of feeling in the party regarding this.

The response is essentially humanitarian. (Of course, there is, as always, a question of how it is to be funded, but given that the US wasted three trillion dollars on prosecuting an illegal war in the Middle East - that is the war in Iraq - on demonstrably false allegations, based upon demonstrably false intelligence - and which the intelligence community itself disowned - it’s clear that the US can well afford such a scheme).

In addition to the other answers, it is possible that they have looked at what happened in the UK when it decided to withdraw free healthcare from undocumented immigrants as part of the "hostile environment" policy. This story describes how a man who had lived legally in the UK for decades was denied cancer treatment for six months because he didn't have the right paperwork.

Edit

US citizens have been detained and deported by immigration because they could not prove their status, so if free universal health care does not apply to illegal immigrants it will also fail to cover some US citizens as well.

In any case, if healthcare is paid through taxes the question should not be "are you a citizen" it should be "are you a taxpayer?". Lots of illegal immigrants pay their taxes so it seems only reasonable to allow them access to the services they are paying for.