For the small values of n I have been able to check, it seems that for $n>3$, there exist whole numbers $x,y$ s.t. $n! = x^2 - y^2$. For example ..
$4! = 5^2 - 1^2$
$5! = 11^2 - 1^2$
$6! = 27^2 - 3^2$
$7! = 71^2 - 1^2$
$8! = 201^2 - 9^2$
$9! = 603^2 - 27^2$
$10! = 1905^2 - 15^2$
$11! = 6318^2 - 18^2$
$12! = 21888^2 - 288^2$
In most of the cases above, the $x$ value is just the next integer larger than $\sqrt{n!}$, though at $n=12$ and $n=17$ it's the one following that. With the tools at hand I've only been able to check this as far as $n=17$.
I expect there's probably already a name for this, but not knowing that name, googling was coming up dry.