The finite simple group $\operatorname{PSp}(4,7)$ has order $138297600 = 11760^2$.

There also seems to be a description of the $q$ such that $\operatorname{PSp}(4,q)$ has square order, see for example here .

Some natural questions:

Which finite simple groups have order $n^2$?

Are there any finite simple groups order $n^k$ for $k \geq 3$?

Is there a classification of finite simple groups of order $n^k$ ($k > 1$)?

For a sporadic simple group $G$, one can check there always exists a prime $p$ such that $p \mid |G|$ and $p^2 \nmid |G|$, so $|G| \neq n^k$ for $k > 1$. The same is also true for $G = \operatorname{Alt}(n)$ by Bertrand's postulate.

The difficult case is then that of the finite simple groups of Lie type.