Two positive integers that differ by 2 can always be written as k+1 and k-1,
k>=2

The difference of their squares is

(k+1)²-(k-1)² = 4k  

4k will be a perfect square if and only if k is a perfect square.

So let k = m²

Then (k+1)²-(k-1)² = 4k = 4m² = (2m)² = n².  Thus n = 2m

I.e., n can be and can only be any even positive integer.

Edwin