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In this paper, we consider the multi-user scheduling problem in millimeter wave (mmWave) video streaming networks, which comprise a streaming server and several users, each requesting a video stream with a different resolution. The main objective is to optimize the long-term average quality of experience (QoE) for all users. We tackle this problem by considering the physical layer characteristics of the mmWave network, including the beam alignment overhead due to pencil-beams. To develop an efficient scheduling policy, we leverage the contextual multi-armed bandit (MAB) models to propose a beam alignment overhead and buffer predictive streaming solution, dubbed B2P-Stream. The proposed B2P-Stream algorithm optimally balances the trade-off between the overhead and users' buffer levels and improves the QoE by reducing the beam alignment overhead for users of higher resolutions. We also provide a theoretical guarantee for our proposed method and prove that it guarantees a sub-linear regret bound. Finally, we examine our proposed framework through extensive simulations. We provide a detailed comparison of the B2P-Stream against uniformly random and Round-robin (RR) policies and show that it outperforms both of them in providing a better QoE and fairness. We also analyze the scalability and robustness of the B2P-Stream algorithm with different network configurations.