The electron energy distribution function (EEDF) is usually assumed to be of the Maxwellian distribution in the fluid model in the simulation of high power microwave breakdown in gas. However, this assumption may lead to some large errors in the simulations. In this paper we compute the non-equilibrium EEDF via solving the Boltzmann equation directly, and incorporate it into the fluid model for argon breakdown. Numerical simulations show that the breakdown time obtained by the fluid model with the non-equilibrium EEDF accords well with the Particle-in-cell-Monte Carlo collision simulation result, while the Maxwellian EEDF has higher energy tail and results in faster breakdown time at low mean electron energy. Based on the non-equilibrium EEDF, the dependence of the breakdown threshold on the pressure predicted by the fluid model accord well with the argon breakdown experimental result.