Let $\mathcal{S}$ be an externally two-valued elementary topos with the axiom of finite choice, i.e. there are only two subobjects of $1$ in $\mathcal{S}$ (up to isomorphism) and any epimorphism with codomain $1$ is a split epimorphism. If $X$ is an object in $\mathcal{S}$ such that there do not exist any morphism $1 \to X$, then $X$ is an initial object in $\mathcal{S}$. Indeed, let $U \rightarrowtail 1$ be the image of the unique morphism $X \to 1$. If $U$ is a terminal object then the unique morphism $X \to 1$ is an epimorphism, in which case the axiom of finite choice implies it has a section, i.e. there exists a morphism $1 \to X$. Thus $U$ is not a terminal object – so it must be an initial object, by two-valuedness. But initial objects in cartesian closed categories are strict, so $X \to U$ must be an isomorphism, and so $X$ must itself be an initial object.
Note that $\mathcal{S}$ need not be well-pointed for the above proof to work: for example, $\mathcal{S}$ could be the topos of simplicial sets. Conversely, any non-degenerate elementary topos with the property that "empty objects are initial" must be externally two-valued and have the axiom of finite choice.