This paper develops energy-efficient hybrid beamforming designs for mmWave multi-user systems where analog precoding is realized by switches and phase shifters such that radio frequency (RF) chain to transmit antenna connections can be switched off for energy saving. By explicitly considering the effect of each connection on the required power for baseband and RF signal processing, we describe the total power consumption in a sparsity form of the analog precoding matrix. However, these sparsity terms and sparsity-modulus constraints of the analog precoding make the system energy-efficiency maximization problem non-convex and challenging to solve. To tackle this problem, we first transform it into a subtractive-form weighted sum rate and power problem. A compressed sensing-based re-weighted quadratic-form relaxation method is employed to deal with the sparsity parts and the sparsity-modulus constraints. We then exploit alternating minimization of the mean-squared error to solve the equivalent problem where the digital precoding vectors and the analog precoding matrix are updated sequentially. The energy efficiency upper bound and a heuristic algorithm are also examined for comparison purposes. Numerical results confirm the superior performances of the proposed algorithm over benchmark energy-efficiency hybrid precoding algorithms and heuristic ones.