I can't figure it out myself. But it seems that the claim must be wrong. The eigenvalues are not integral. For example, with $n=4$ the matrix is
$$
A=\begin{pmatrix}
0 & 4 & 0 & 0\cr
1 & 0 & 3 & 0\cr
0 & 2 & 0 & 2 \cr
0 & 0 & 3 & 0
\end{pmatrix}.
$$

The characteristic polynomial of this matrix is 
$\chi (t)=t^4-16 t^2 +24$, which has no integral roots.
Am I overlooking something ?