Topologically each point on the edge of a square can be mapped uniquelly (including the corners?) to a point on the edge of a circle.
It seems it might be possible to deform in 3D space the square and the disc so that they can be joined at the edge without any gaps. A colleague managed to do that in the real world with paper objects. Obviously this is not a precise result, so it may be wrong.
As a commenter said, the deformation should be isometric.
Can you make Mathematica solve this problem and generate a 3D surface showing a isometrically deformed square and disc joined at the edge?
Pictures taken from http://www.unitaryflow.com/2015/03/round-squares-exist.html