Efficient mitigation of power amplifier (PA) nonlinear distortion in multi-user hybrid precoding based broadband mmWave systems is an open research problem. In this article, we first carry out detailed signal and distortion modeling in broadband multi-user hybrid MIMO systems, with a bank of nonlinear PAs in each subarray, while also take the inevitable crosstalk between the antenna/PA branches into account. Building on the derived models, we then propose an efficient closed-loop (CL) digital predistortion (DPD) solution that requires only a single-input DPD unit per transmit chain or subarray, despite crosstalk, providing substantial complexity-benefit compared to the state-of-the art multi-dimensional DPD solutions. We show that under spatially correlated multipath propagation, each DPD unit can provide linearization towards every intended user, or more generally, towards all spatial directions where coherent propagation is taking place, and that the proposed CL DPD system is robust against crosstalk}. Extensive numerical results building on practical measurement-based mmWave PA models are provided, demonstrating and verifying the excellent linearization performance of the proposed DPD system in different evaluation scenarios.