Polyelectrolyte gels are known to exhibit strong response to A changes in salt concentration, an effect attributed to osmotic effects. Traditional theories are insufficient to describe all the properties observed in experiments; namely, reentrant swelling effects at high salt concentrations and complete polyelectrolyte collapse at high salt valencies. We incorporate liquid-state integral equation theories to account for the many-body correlations between finite-sized ions in a strongly charged polyelectrolyte gel. This represents a theory for such systems that goes well beyond well-known Debye-Huckel (DH) and Poisson-Boltzmann (PB) theories. We demonstrate that charge correlations give rise to enhanced gel deswelling, a first-order collapse to a highly correlated state at large valencies, and counterion size-dependent reentrant swelling. Such ion-specific effects provide routes for practical control of stimuli-responsive polymeric materials based on polyelectrolyte gels, and deeper understanding of biological systems.