This paper studies the cross-layer challenges and performance of Hybrid Beamforming (HBF) and Multi-User Multiple-Input Multiple-Output (MU-MIMO) in 5G millimeter wave (mmWave) cellular networks with full-stack TCP/IP traffic and MAC scheduling. While previous research on HBF and MU-MIMO has focused on link-level analysis of full-buffer transmissions, this work reveals the interplay between HBF techniques and the higher layers of the protocol stack. To this aim, prior work on the full-stack evaluation of mmWave cellular networks has been extended by including the modeling of MU-MIMO and HBF. Our results reveal novel relations between the networking layers and the HBF MU-MIMO performance at the physical layer. Particularly, throughput can be increased in 5G networks by means of Space Division Multiple Access (SDMA). However, in order to achieve such benefits it is necessary to take into account certain trade-offs and the implementation complexity of a full-stack HBF solution.