Assume all fixed points are on the boundary. For each fixed point you can find an invariant neighborhood. (You can assume that the $Z/nZ$-action fixes some Riemannian metric. Then any $\epsilon$-ball around the fixed point is invariant.) These sets are homeomorphic to a half-ball.
The complement of the union of these half-balls (over all fixed points) is homeomorphic to a closed ball. The action on this closed ball does not have fixed points, contradicting Brouwer's theorem.