We study experimentally and numerically a new dynamical regime in the operation of semiconductor ring lasers (SRLs) subject to delayed optical feedback. When employing an asymmetric feedback scheme, we find experimentally that the SRL can show square-wave intensity oscillations with a 50 % duty cycle. In this scheme, where the output in one direction is delay-coupled to the other direction but not vice versa, the laser switches regularly between the clockwise (CW) and counter-clockwise (CCW) propagating modes. The measured period of the square-waves is slightly longer than twice the roundtrip time in the external cavity. We analyze the regularity and the shape of the square-waves as a function of the pumping current and the feedback strength. For higher pump currents on the SRL,the output displays stochastic mode hopping between the square waves attractor and stable unidirectional operation in the CW mode. To understand the origin of this dynamical regime, we rely on numerical simulations based on the Lang-Kobayashi equations. We demonstrate a novel mechanism leading to square wave oscillations based on the cross-feedback overcoming backscattering asymmetries present in the device's structure. Our numerical results are in close agreement with the experimental ones.