This is false for $n=1.$ The mapping class group of the torus is $SL(2, Z),$ of which the homeomorphisms you describe are but a small part - the parabolic matrices $\begin{pmatrix}1 & n\\ 0 &1\end{pmatrix},$ unless I am very confused. I cautiously believe the statement is true for $n=2,$ by 

<cite authors="Allen Hatcher" mrnumber="420620" cite="_Topology_ **15** (1976), no. 4, 343--347">_Allen Hatcher_, MR 420620 [**Homeomorphisms of sufficiently large $P^{2}$-irreducible $3$-manifolds**](http://www.ams.org/mathscinet-getitem?mr=420620), _Topology_ **15** (1976), no. 4, 343--347.</cite>