As a non-mathematician, for a 2d circle on a flat plane:

- The edge of a circle is curved (convex)

- It does not matter how small a piece you cut of the edge, there will always remain a convex curve.

- The edge of a square has straight edges, so you cannot put this piece at the edge. This means the old outside edge will have to be on the inside.

- You cannot fit the convex edges to each other or to a straight edge.

- Cutting a concave edge from the inside of the circle to fit the convex edge from the outside to will not help as it will produce a new convex edge.

Ergo: there is no place to put the convex outside edge of the circle, so you cannot turn it into a square.