We report results of calculations of the second through sixth virial coefficients for four prototype Lennard-Jones (LJ) mixtures that have been the subject of previous studies in the literature. Values are reported for temperatures ranging from T=0.6 to T=10.0, where here the temperature is given units of the LJ energy parameter of one of the components. Thermodynamic stability of the mixtures is studied using the virial equation of state (VEOS) with the calculated coefficients, with particular focus on characterizing the vapor-liquid critical behavior of the mixtures. For three of the mixtures, vapor-liquid coexistence and critical data are available for comparison at only one temperature, while for the fourth we can compare to a critical line. We find that the VEOS provides a useful indication of the presence and location of critical behavior, although in some situations we find need to consider "near-miss" critical behavior, where the classical conditions of criticality are nearly but not exactly satisfied.