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:
$|{\rm PSL}(2,7)| + 1 = 13^2$,
$|{\rm A}_6| + 1 = 19^2$,
$|{\rm M}_{11}| + 1 = 89^2$,
$|{\rm PSU}(4,2)| + 1 = 161^2$,
$|{\rm J}_1| + 1 = 419^2$