Let me apologize in advance as this is possibly an extremely stupid question: can one prove or disprove the existence of a bijection from the plane to itself, such that the image of any circle becomes a square? Or, more generally, are there any shapes other than a square such that a bijection does exist? (obviously, a linear map sends a circle to an ellipse of fixed dimensions and orientation)
There is no such bijection.
To see this, imagine four circles all tangent to some line at some point $p$, but all of different radii, so that any two of them intersect only at the point $p$. (E.g., any four circles from this picture.) Under your hypothetical bijection, these four circles would map to four squares, any two of which have exactly one point in common, the same point for any two of them. You can easily convince yourself that no collection of four squares has this property.