SOLUTION
Now looking at the cube below,
The sides marked with a blue pen and labelled as "a" are equal in length, but the side marked as "h" is longer because it is acting as the hypotenuse of the right triangle so formed. Hence the sides of the quadrilateral formed from the intersection is a Rectangle, since only two sides are equal.
So,
This intersection described above is a Rectangle. The diagonal across the top and bottom square bases is Greater than the edges of the cube. Because the diagonal across a square acts like the Hypotenuse of an isosceles right triangle.