Hydrophobically modified copolymers provide a versatile platform of hydrogel materials for diverse applications, but the influence of salts on the swelling and material properties of this class of hydrogels has not been extensively studied. Here, we investigate model hydrogels with three different sodium salts with anions chosen from the classic Hofmeister series to determine how these counterions influence the swelling and mechanical properties of neutral hydrogels. The gel chosen was based on a statistical copolymer of dimethylacrylamide and 2-(N-ethylperfluorooctane sulfonamido) ethyl acrylate (FOSA). Our measurements utilize a quartz crystal microbalance with dissipation (QCM-D) to quantify both swelling and rheological properties of these gels. We find that a 1 mol/L solution of Na2SO4, corresponding to a kosmotropic anion, leads to nearly a 2.6-fold gel deswelling and correspondingly, the complex modulus increases by an order of magnitude under these solution conditions. In contrast, an initial increase in swelling and then a swelling maximum is observed for a 0.02 mol/L concentration in the case of a chaotropic anion, NaClO4, but the changes in the degree of gel swelling in this system are not directly correlated with changes in the gel shear modulus. The addition of NaBr, an anion salt closer to the middle of the chaotropic to kosmotropic range, leads to hydrogel deswelling where the degree of deswelling and the shear modulus are both nearly independent of salt concentration. Overall, the observed trends are broadly consistent with more kosmotropic ions causing diminished solubility ("salting out") and strongly chaotropic ions causing improved solubility ("salting in"), a trend characteristic of the Hoffmeister series governing the solubility of many proteins and synthetic water-soluble polymers, but trends in the shear stiffness with gel swelling are clearly different from those normally observed in chemically cross-linked gels and are correspondingly difficult to interpret. The salt specificity of swelling and mechanical properties of nonionic hydrogels is important for any potential application in which a wide range of salt concentrations and types are encountered.