For the uniqueness part, other than using maximum principle as Thomas said, you can also prove in this way: For $u$ satisfying $\tag{1}u''+qu=0$ with $q\leq 0$ and $u(0)=u(1)=0$, we must have $u\equiv 0$. To see this, multiply $(1)$ by $u$ and integrate it over $[0,1]$, we get $\int_0^1uu''+\int_0^1qu^2=0.$ Integration by parts, we have $\big[uu'\big]_0^1-\int_0^1(u')^2+\int_0^1qu^2=0.$ Since $u(0)=u(1)=0$ by assumption, we have $-\int_0^1(u')^2+\int_0^1qu^2=0.$Left hand side is nonpositive since $q\leq 0$, which implies that $u'\equiv 0$, i.e. $u\equiv c$ for some constant $c$. Since $u(0)=u(1)=0$, $c=0$, i.e. $u\equiv 0$ as required.