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I'm looking for a symbol like this

The only problem was finding a suitably small square symbol, which can be found in mathabx and imported.

A few tricks can make for defining also the corresponding left arrow avoiding code duplication. The square is positioned so that the distance from the arrow end to it is half than the distance to the arrow tip; this is of course reversed for the left arrow. You can vary this choice by changing 12 into, say 1.52.5 or whatever you think fit.

documentclassarticle
usepackageamsmath,amssymb
usepackagemarvosym

DeclareFontFamilyUmathb
DeclareFontShapeUmathbmn
 <-5.5> mathb5
 <5.5-6.5> mathb6
 <6.5-7.5> mathb7
 <7.5-8.5> mathb8
 <8.5-9.5> mathb9
 <9.5-11> mathb10
 <11-> mathb12

DeclareSymbolFontmathbUmathbmn
DeclareMathSymbolabxsmallsquare2mathb"05

makeatletter
DeclareRobustCommandsqrightarrow%
 mathrelmathpalettesq@arrow12rightarrow%

DeclareRobustCommandsqleftarrow%
 mathrelmathpalettesq@arrow21leftarrow%

newcommandsq@arrow[2]sq@@arrow#1#2
newcommandsq@@arrow[4]%
 ooalign%
 hskip 0pt plus #2fil
 $m@th#1abxsmallsquare$%
 hskip 0pt plus #3filcr
 $m@th#1#4$cr%

makeatother

begindocument

$asqrightarrow b sqleftarrow c$

$scriptstyle asqrightarrow b sqleftarrow c$

$scriptscriptstyle asqrightarrow b sqleftarrow c$

enddocument

Two variant solution based on the stackinset command from stackengine and relsize or rotating:

documentclass[border = 2pt]standalone

usepackagestackengine %
usepackageamssymb, relsize, rotating
usepackageold-arrows 
newcommandsqarrowstackMathmathrelstackinsetc0exc0.14exmathsmallermathsmallerBox ―――→ 
newcommandvarsqarrowstackMathmathrelstackinsetc0exc0.12exrotatebox45$ diamond $ ―――→ 
begindocument

 $ A sqarrow B$ quad

$ A varsqarrow B$

enddocument