In this paper, we design an uplink/downlink channel estimation method for millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) systems and investigate the impact of beam squint effect that accompanies large array configuration. Specifically, we adopt the off-grid sparse Bayesian learning (SBL) that directly works on the continuous angle-delay parameter domain and avoids the grid mismatch problem. Hence, the proposed method achieves good channel estimation accuracy and handles the wideband direction of arrival (DOA) estimation problem for mmWave massive MIMO communications, where beam squint effect was previously ignored by many existing literatures. The Cramér-Rao bound for unknown parameters is derived to make the proposed study complete. More importantly, a much simplified downlink channel estimation scheme is designed with the aid of angle-delay reciprocity, which significantly reduces training and feedback overhead. The simulation results are provided to demonstrate the superior performance of the proposed method over existing ones.