Is this non-separable utility?
You are right, these preferences are separable. A way to see it is to notice that
\begin{equation*}
U(c,n) \geq U(c',n) \Rightarrow U(c,n') \geq U(c',n')
\end{equation*}
for any $n,n'$, and 
\begin{equation*}
U(c,n) \geq U(c,n') \Rightarrow U(c',n) \geq U(c',n')
\end{equation*}
for any $c,c'$.
They are even additively separable, as your log-transformation shows.