I've attached adrawing to show you how this construction can be done:

1. The concentric circles are the paths on which the vertices of the square are moving during the rotation.

2. Construct a line perpendicular to OA.

3. Where this line intersect the circle passing through A the image of A, A', is located.

4. Do the same procedure for the last 3 vertices.

5. Connect all points A', B', C' and D'. Check if you actually got a square.