What is the most simple way to find an area of two squares' intersection?

Find it as intersection of two arbitrary convex polygons, otherwise you'll bury yourself in special cases.

To do so get all points of one polygon inside the other, all points of the other polygon inside the first one, and also all their intersections by non-parallel sides.

Then find area of their convex hull. Angle/distance sorting step is enough to find that hull.