The minimum number is 14 squares.
A solution with 14 squares it to place four 7x7 squares, four 6x6 squares, and four 5x5 squares touching the four corners of the boards, plus two 4x4 squares touching two opposite corners.
To show that 14 is optimal, consider the 28 segments that touch the outer edge of the board but don't lie on it. Any square covers a maximum of two of these segments: no square can reach two segments adjoining two opposite sides, if it touches two segments adjoining the same side, it can't reach any other segments. So, 14 squares are needed.
This also implies that in any 14-square solution, every square must touch the edge of the board, and the 8x8 square cannot be used.