Boundary conditions for fluid-dynamic variables of strongly nonequilibrium multicomponent gas mixture flows in slip regime are derived systematically by two different approaches. The flow is described in the framework of the state-to-state model for coupled detailed vibrational and chemical kinetics. The specular-diffusive model is applied for particles interaction with the solid wall, and the surface is assumed to be partially catalytic with possible state-specific chemical reactions and vibrational energy transitions described by a simple model based on the analogy with recombination coefficients. The first theoretical approach uses the technique proposed by Grad whereas the second one is based on the kinetic boundary condition. It is shown that for the Maxwell scattering kernel the two approaches are equivalent; at the same time, the approach based on the kinetic boundary condition provides more rigorous mathematical description of the problem and can be easily generalized for other scattering kernels and gas-surface interaction models accounting for surface inner geometry. The resulting boundary conditions are expressed in terms of state-specific transport coefficients: thermal conductivity, multicomponent diffusion of vibrational states, thermal diffusion, shear and bulk viscosity, and relaxation pressure. The dependence of the boundary conditions on the normal mean stress is obtained for the first time. Under thermal equilibrium conditions, the derived expressions reduce to known relations obtained earlier in the one-temperature approach.