Well, recognize that what you want to construct is an arithmetic progression of three squares, where the middle square is and the common difference is (reversing the order to indicate ). There is a parametric formula for progressions of three squares that says: for , giving a common difference of . (Click here for a wonderful proof that this captures all of them.) So, your question becomes:
Find positive integers with satisfying
You might start by isolating in the first equation, subbing it into the second, and attempting to isolate . Not sure if this is possible directly because this gives you a quartic polynomial on , but if you can, then the problem will reduce to proving when or whether your expression evaluates to an integer. I know I'm being lazy, but that should give you something to chew on.