(In reply to computer solution
Nice analysis, Charlie! A nice problem, but I suspected a misprint since the first sentence did not seem to describe the diagram. There were "the first 8 square numbers (one twice)" (49 in twice). BUT "and all of the primes (except one) below 65" was amiss since neither 13 nor 17 appear.
The final "food for thought" again made me wonder since there are 8 squares below 65, and 18 primes below 65 -- for a total of 26 cells, not a 5x5 matrix. Perhaps this puzzle should have been saved for Friday the 13th??