The board will be a $3 \times 3$ one and will look as follows:

We know that to form a square/rectangle we need 2 horizontal and 2 vertical lines.

The number of ways in which 2 vertical lines may be chosen from 4 = $4C2 = \frac{(4 \times 3)}{2} = 6$

Similarly, number of ways in which 2 horizontal lines may be chosen from 4 = $4C2 = \frac{(4 \times 3)}{2} = 6$

Hence, the maximum number of squares and rectangles that can be formed = $6 \times 6 = 36$