The equations of state and band-gap closures for $\mathrm{Pb}{\mathrm{Cl}}_{2}$ and $\mathrm{Sn}{\mathrm{Cl}}_{2}$ were studied using both experimental and theoretical methods. We measured the volume of both materials to a maximum pressure of 70 GPa using synchrotron-based angle-dispersive powder x-ray diffraction. The lattice parameters for both compounds showed anomalous changes between 16--32 GPa, providing evidence of a phase transition from the cotunnite structure to the related ${\mathrm{Co}}_{2}\mathrm{Si}$ structure, in contrast to the postcotunnite structure as previously suggested. First-principles calculations confirm this finding and predict a second phase transition to a ${\mathrm{Co}}_{2}\text{Si-like}$ structure between 75-- 110 GPa in $\mathrm{Pb}{\mathrm{Cl}}_{2}$ and 60--75 GPa in $\mathrm{Sn}{\mathrm{Cl}}_{2}$. Band gaps were measured under compression to \ensuremath{\sim}70 GPa for $\mathrm{Pb}{\mathrm{Cl}}_{2}$ and \ensuremath{\sim}66 GPa for $\mathrm{Sn}{\mathrm{Cl}}_{2}$ and calculated up to 200 GPa for $\mathrm{Pb}{\mathrm{Cl}}_{2}$ and 120 GPa for $\mathrm{Sn}{\mathrm{Cl}}_{2}$. We find an excellent agreement between our experimental and theoretical results when using the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional, which suggests that this functional could reliably be used to calculate the band gap of similar $A{X}_{2}$ compounds. Experimental and calculated band-gap results show discontinuous decreases in the band gap corresponding to phase changes to higher-coordinated crystal structures, giving insight into the relationship between interatomic geometry and metallicity.