The roots are always simple, this follows without further reasoning from the old paper On the irreducibility of certain trinomials and quadrinomials by Ljunggren. Set $f(z)=z^n-1-z^p-z^q$. Then Ljunggren proves (Theorem 1) that $f(z)=Q(z)g(z)$, where the roots of $Q$ are roots of unity (so $Q=1$ if there are no such roots), and where $g(z)\in\mathbb Q[z]$ is irreducible none of its roots is a root of unity. As an irreducible polynomial, $g$ has no multiple roots. Furthermore (Theorem 2), Ljunggren shows that $Q$ has simple roots either, which settles the question.
Remark: In case Ljunggren's paper is not accessible, this review at zbMATH contains the full statement of Ljunggren's results needed here.