We consider the hard-core model on a finite square grid graph with stochastic Glauber dynamics parametrized by the inverse temperature $β$. We investigate how the transition between its two maximum-occupancy configurations takes place in the low-temperature regime $β\to\infty$ in the case of periodic boundary conditions. The hard-core constraints and the grid symmetry make the structure of the critical configurations, also known as essential saddles, for this transition very rich and complex. We provide a comprehensive geometrical characterization of the set of critical configurations that are asymptotically visited with probability one. In particular, we develop a novel isoperimetric inequality for hard-core configurations with a fixed number of particles and we show how not only their size but also their shape determines the characterization of the saddles.