  Given a group $G$ and a subgroup $H$, we let $\mathcal{O}_G(H)$ denote the
lattice of subgroups of $G$ containing $H$. This paper provides a
classification of the subgroups $H$ of $G$ such that $\mathcal{O}_{G}(H)$ is
Boolean of rank at least $3$, when $G$ is a finite alternating or symmetric
group. Besides some sporadic examples and some twisted versions, there are two
different types of such lattices. One type arises by taking stabilizers of
chains of regular partitions, and the other type arises by taking stabilizers
of chains of regular product structures. As an application, we prove in this
case a conjecture on Boolean overgroup lattices, related to the dual Ore's
theorem and to a problem of Kenneth Brown.
