This paper proposes nonlinear least square error (LSE) precoders for multiuser MIMO broadcast channels. The LSE precoders are designed such that the discrete output signals are from a predefined set. This predefined set allows us to model several signal constraints such as peak power constraint, constant envelope, and discrete constellations. We study the large-system performance of these precoders via the replica method from statistical physics, and derive a closed-form expression for the asymptotic distortion. Our results demonstrate that an LSE precoder with the output peak-to-average power ratio of 3 dB can perform similar to the regularized zero forcing (RZF) precoder. As the peak-to-average power ratio reduces to one, the constant envelope precoder is recovered. The investigations show that the performance of the RZF precoder is achieved by a constant envelope precoder with 20% additional transmit antennas. For M-phase shift keying constellations, our analysis gives a lower bound on the asymptotic distortion which is tight for moderate antenna-to-user ratios and deviates as the ratio grows. We improve this bound by deriving the replica solution under one-step of replica symmetry breaking. Our numerical investigations for this case show that the bound is tight for antenna-to-user ratios less than 5.