All sorts - read the cited wikipedia articles for starters - the most important being that it's a small finite number that grants exceptional packing efficiency in the Leech lattice, which is then the basis for the CFT tie-in.
Apart from anything else, this sort of thing is just beautiful mathematically.
for the general case, the article states there are not infinite such cases:
"A mathematician might state this property of the number 24 as follows:
If m is relatively prime to 24, then m^2 is congruent to 1 modulo 24.
One might ask if any numbers other than 24 have this property. The answer is “yes”, but the only other numbers that exhibit this property are 12, 8, 6, 4, 3, 2 and 1; in other words, the factors of 24."
Surely there are infinite such cases for square primes.
prime^2 = ( n * m ) + c
Where n and c are constants.