The hadron-resonance gas (HRG) model with the mass-proportional eigenvolume (EV) corrections is employed to fit the hadron yield data of the NA49 collaboration for central Pb+Pb collisions at $\sqrt{s_{{\rm NN}}} = 6.3, 7.6, 8.8, 12.3,$ and $17.3$ GeV, the hadron midrapidity yield data of the STAR collaboration for Au+Au collisions at $\sqrt{s{\rm NN}} = 200$ GeV, and the hadron midrapidity yield data of the ALICE collaboration for Pb+Pb collisions at $\sqrt{s_{\rm NN}} = 2760$ GeV. At given bombarding energy, for a given set of radii, the EV HRG model fits do not just yield a single $T-\mu_B$ pair, but a whole range of $T-\mu_B$ pairs, each with similarly good fit quality. These pairs form a valley in the $T-\mu_B$ plane along a line of nearly constant entropy per baryon, $S/A$, which increases nearly linearly with bombarding energy $E_{\rm lab}$. The entropy per baryon values extracted from the data at the different energies are a robust observable: it is almost independent of the details of the modeling of the eigenvolume interactions and of the specific $T-\mu_B$ values obtained. These results show that the extraction of the chemical freeze-out temperature and chemical potential is extremely sensitive to the modeling of the short-range repulsion between the hadrons. This implies that the ideal point-particle HRG values are not unique. The wide range of the extracted $T$ and $\mu_B$ values suggested by the eigenvolume HRG fits, as well as the approximately constant $S/A$ at freeze-out, are consistent with a non-equilibrium scenario of continuous freeze-out, where hadrons can be chemically frozen-out throughout the extended space-time regions during the evolution of the system. Even when the EV HRG fits are restricted to modest temperatures suggested by lattice QCD, the strong systematic effects of EV interactions are observed.