We extend the SAFT-VRQ Mie equation of state, previously developed for pure fluids [Aasen et al., J. Chem. Phys. 151, 064508 (2019)], to fluid mixtures of particles interacting via Mie potentials with Feynman-Hibbs quantum corrections of first (Mie-FH1) or second order (Mie-FH2). This is done using a third-order Barker-Henderson expansion of the Helmholtz energy from a non-additive hard-sphere reference system. We survey existing experimental measurements and ab initio calculations of thermodynamic properties of mixtures of neon, helium, deuterium and hydrogen, and use them to optimize the Mie-FH1 and Mie-FH2 force fields for binary interactions. Simulations employing the optimized force fields are shown to follow the experimental results closely over the entire phase envelopes. SAFT-VRQ Mie reproduces results from simulations employing these force fields, with the exception of near-critical states for mixtures containing helium. This breakdown is explained in terms of the extremely low dispersive energy of helium, and the challenges inherent in current implementations of the Barker-Henderson expansion for mixtures. The interaction parameters of two cubic equations of state (SRK, PR) are also fitted to experiments, and used as performance benchmarks. There are large gaps in the ranges and properties that have been experimentally measured for these systems, making the force fields presented especially useful.