The more natural conjecture, that <i>every</i> even number arises as the difference of two primes, <a href="https://math.stackexchange.com/q/211438">was asked on math.SE years ago</a>. That question remains open, and I have a hard time imagining that restricting to squares makes the problem any more tractable.

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<b>EDIT:</b> The conjecture that every even number is the difference of two primes is apparently sometimes called <a href="https://mathoverflow.net/a/111208">Maillet's conjecture</a>.