Proving that $b^6-2a^3b^3-2b^3+a^6-2a^3+1$ is never a nontrivial integer square?

Question

I'm trying to prove that if $a,b$ are integers, and $b^6-2a^3b^3-2b^3+a^6-2a^3+1$ is a square [integer], then $ab=0$.

What general tools are available to attack such a problem?