I hate any bodily touch!!

The number of ways of selecting two squares that share a vertical edge is $8 \times 7 = 56$ . Similarly there are 56 ways of selecting two squares that share a horizontal edge. There are $7 \times 7 = 49$ ways of selecting two squares that share a vertex in a bottom-left to top-right configuration and similarly 49 ways that share a vertex in the top-left to bottom-right configuration.

Hence the desired answer is $\binom{64}{2} - 2(56 + 49) = 32\times63 - 210 = 2016-210 = \boxed{1806}.$

We can split this problem into 3 cases:

1. The first square is on a corner

2. The first square is on an edge (but not a corner)

3. The first square is in the middle (not an edge or corner)

There are 4 ways to choose a corner as the first square, and 60 ways to choose the second square afterwards. We can multiply the amounts together. $4 \times 60$ = 240

There are 24 squares on the edge which we can choose as the first square, and 58 squares to choose as the second square afterwards. $24 \times 58$ = 1392

There are 36 squares in the middle which we can choose as the first square, and 55 squares to choose as the second square. $36 \times 55$ = 1980

We can add the 3 numbers we have gotten together. $240 + 1392 + 1980$ = 3612

However, $\frac{1}{2!}$ of our 3612 ways to choose the two squares are repeats, since the order of the squares doesn't matter. Therefore, we must divide 3612 by 2!, which is just 2, to get a final answer of $\boxed {1806}$