The nonlinear Langevin equation theory of segmental relaxation, elasticity, and nonlinear mechanical response of deformed polymer glasses with aging and mechanical rejuvenation processes taken into account is applied to study material response under a constant strain rate deformation. In the postyield softening regime, the amplitude of the stress overshoot feature, and its breadth in strain, are predicted to be positively correlated with the mechanically induced disordering process. The key physics is the increase of the density fluctuation amplitude due to mechanically generated disorder (rejuvenation) which reduces the elastic modulus and speeds up relaxation beyond the effects of the landscape tilting mechanism. Detailed numerical calculations reveal that the emergence of strain softening is not directly tied to a difference between the initial and steady plastic flow states, but rather on whether there exists a rejuvenation-dominated process during deformation. Calculations suggest a roughly linear relation between the strain softening amplitude (SSA) and the amount of rejuvenation as quantified by variation of the density fluctuation amplitude. The dependences of the yield stress and strain, steady state flow stress, and SSA on deformation rate, temperature, preaging time, and also two distinct thermal history protocols are investigated in detail for PMMA glass. Overall, good agreement between theory and experiment is found.