We investigate lysozyme-lysozyme and lysozyme-salt interactions in electrolyte solutions using a molecular-thermodynamic model. An equation of state based on the statistical mechanical perturbation theory is applied to describe the interactions. The perturbation term includes a new square-well potential of mean force, which implies the information about the lysozyme surface and salt type. The attractive energy of the potential of mean force is correlated with experimental cloud-point temperatures of lysoyme in various solution conditions. The same attractive energy is used to predict osmotic pressure of a given system with no additional parameters. The new potential shows a satisfactory improvement in understanding the interactions between lysozymes in aqueous salt solutions.