Using the formula for the pdf of the Irwin-Hall distribution one gets $$\frac{\sqrt{n}}{(n-1)!}\sum_{k=0}^{\lfloor \frac{n}{2}\rfloor}(-1)^{k}{n \choose k}\left(n-2k\right)^{n-1}$$
It's fairly straightforward to see why, imagine you're drawing random point in your cube, how many fall within a distance $\epsilon$ of the hyperplane? Now consider that the thickness of the slice is $2\sqrt{n}\epsilon$. Take the limit as $\epsilon \rightarrow 0$
This paper proposes an algorithm for a slight generalisation