We study theoretically and experimentally the combined effects of polarization-selective optical feedback and of crossed-polarization reinjection in a vertical-cavity surface-emitting laser. We show that the application of polarization-selective optical feedback, which induces an effective dichroism in the system, allows one to generate a robust and regular square-wave output signal in each polarization component. The period of the square-wave signal is determined by twice the reinjection delay. We analyze the regularity of the induced modulation as a function of laser bias current, dichroism, and of the levels and delays of reinjection and feedback, thus revealing the robustness of the square-wave emission in the parameter space. We also show that the feedback is more effective when acting on the long-wavelength polarization mode, which can be traced to the asymmetry of the bistable region of the vertical-cavity surface-emitting lasers for low dichroism.