In "Subgroups of Graph Groups", 1987, J. Alg., Droms proved that all the subgroups of a right-angled Artin group (RAAG) defined by a finite simplicial graph $Γ$ are themselves RAAGs if, and only if, $Γ$ has no induced square graph nor line-graph of length $3$. The present work provides a similar result for specific normal subgroups of RAAGs, called Bestvina-Brady groups: We characterize those graphs in which every subgroup of such a group is itself a RAAG. In turn, we confirm several Galois theoretic conjectures for the pro-$p$ completions of these groups.