Orthogonal time frequency space (OTFS) modulation and massive multi-input multi-output (MIMO) are promising technologies for next generation wireless communication systems for their abilities to counteract the issue of high mobility with large Doppler spread and mitigate the channel path attenuation, respectively. The natural integration of massive MIMO with OTFS in millimeter-wave systems can improve communication data rate and enhance the spectral efficiency. However, when transmitting wideband signals with large-scale arrays, the beam squint effect may occur, causing discrepancies in beam directions across subcarriers in multi-carrier systems. Moreover, the high-mobility wideband millimeter wave communications can induce the Doppler squint effect, leading to different Doppler shifts among the subcarriers. Both beam squint effect and Doppler squint effect (denoted as doubly squint effect) can degrade communication performance significantly. In this paper, we present an efficient channel estimation and hybrid precoding scheme to address the doubly squint effect in massive MIMO-OTFS systems. We first characterize the wideband channel model and the input-output relationship for massive MIMO-OTFS transmission considering doubly squint effect. We then mathematically derive the impact of channel parameters on chirp pilots under the doubly squint effect. Additionally, we develop a peak-index-based channel estimation scheme. By leveraging the results from channel estimation, we propose a hybrid precoding method to mitigate the doubly squint effect in downlink transmission scenarios. Finally, simulation results validate the effectiveness of our proposed scheme and show its superiority over the existing schemes.