Bijection from the plane to itself that sends circles to squares

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.

The problem is that two distinct squares can have more than two common points (easy to make an example), and under such a bijection these squares would have to go to two distinct circles with more than two common points - an impossibility.