The rubber elasticity theory has been lengthily applied to several polymeric hydrogel substances and upgraded from idealistic models to consider imperfections in the polymer network. The theory relies solely on hyperelastic material models in order to provide a description of the elastic polymer network. While this is also applicable to polymer gels, such hydrogels are rather characterized by their water content and visco-elastic mechanical properties. In this work, we applied rubber elasticity constitutive models through hyperelastic parameter identification of hydrogels based on their stress–strain response to compression. We further performed swelling experiments and determined the intrinsic properties, i.e., density, of the specimens and their components. Additionally, we estimated their equilibrium swelling and employed it in the swelling-equilibrium theory in order to determine the polymer–solvent interaction parameter of each hydrogel with regard to cross-linking. Our results show that the average mesh size obtained from the rubber elasticity theory can be regarded as a concentration-dependent characteristic length of the hydrogel’s network and couples the non-linear elastic response to the specimens’ inherent visco-elasticity through hysteresis as a quantifier of energy dissipation under large deformation.