My question is a refinement of this one about 'efficient' construction of square elements: If the word problem for a (finitely generated, finitely presented) group is decidable, must the 'square problem' (given an element $g$ of $G$, is there an element $h$ with $g=h^2$?) also be decidable? If not, how 'nice' can $G$ be while still having an undecidable square problem? For instance, can $G$ be automatic? (It feels like there should be an argument based on the Dehn function that precludes this, but I'm not immediately seeing it.) Could it even be hyperbolic?