Is it not zero whenever $n \geq 2$?  Let $A$ be a $n \times n$ permutation matrix with determinant $-1$ (which requires $n \geq 2$).  Then the uniform distribution of a random $n \times n$ $(0,1)$-matrix $X$ is the same as the distribution of $XA$.  The permanent is multiplicative, hence Per$(AX)=$Per$(A)$Per$(X)=-$Per$(X)$.  Hence the probability of Per$(X)=x$ is the same as the probability of Per$(X)=-x$.