Van der Waals theory predicts that the attractive contribution to the compressibility factor is proportional to the inverse of temperature and volume. However, there are deviations from this linear behavior caused by both molecular attractions and molecular repulsions. Here, deviations from linearity due to repulsions are predicted by first order perturbation theory and deviations due to attractions are predicted by an approximate solution to the 3D Ising problem. The close agreement between U/Umean-field for lattice and off-lattice systems is used to derive a new analytic EOS for off-lattice systems which shows very good agreement with simulation data for square-well (SW) and Lennard-Jones (LJ) fluids. While the structures of lattice and off-lattice systems are very different, we show that the non-randomness due to molecular attractions of lattice and off-lattice fluids are fundamentally the same. Because the lattice theory can be generalized to multi-component mixtures without additional assumptions, the model can be extended rigorously to mixtures, i.e., no empirical mixing rules are introduced. The EOS is applied to mixtures of hard-sphere (HS) and square-well (SW) molecules, as well as to SW mixtures with different well depths. The application to highly non-random mixtures of HS and SW mixtures shows very good agreement in internal energy results and fair agreement in compressibility factor calculations. A comparison to the second-order perturbation term for mixtures shows good agreement whereas van der Waals one fluid theory shows large deviations. Second virial coefficients for SW and LJ molecules also are described very accurately with the new approach.