 Solution

For $n=0$ the proposed wavefunction vanishes identically and cannot be normalized. A negative integer gives $\sin (-|n|\pi x/L)=-\sin (|n|\pi x/L)$, which differs from the corresponding positive-integer wavefunction only by a physically irrelevant global phase. Both have the same positive energy, proportional to $n^2$. Thus zero is inadmissible and negative labels are redundant. A negative integer is not forbidden by a separate negative-energy argument.