$abcd$ is a square. how many squares have two or more vertices in the set $\{a, b, c, d\}$?

There are 4 squares having a side corresponding to a side of square abcd.

There are 4 squares having a side corresponding to the diagonals of square abcd.

And then the original square

Therefore, there are 4 + 4 + 1 = 9 squares having two or more vertices in the set {a, b, c, d}.