For abelian simple groups your question is merely a disguised form of
"enumerate the Mersenne primes".
The smallest examples for nonabelian simple groups are as follows:

  - $|{\rm PSL}(2,7)| + 1 = 13^2$,

  - $|{\rm A}_6| + 1 = 19^2$,

  - $|{\rm M}_{11}| + 1 = 89^2$,

  - $|{\rm PSU}(4,2)| + 1 = 161^2$,

  - $|{\rm J}_1| + 1 = 419^2$