Non-linear transceivers and non-reciprocity of downlink and uplink channels are two major challenges in the deployment of massive multiple-input-multiple-output (MIMO) systems. We consider an over-the-air (OTA) approach for digital pre-distortion (DPD) and reciprocity calibration to jointly address these issues. In particular, we consider a memory-less non-linearity model for the base station (BS) transmitters, and we propose a method to perform both linearization and reciprocity calibration based on mutual coupling OTA measurements between BS antennas. We show that, by using only the OTA-based data, we can linearize the transmitters and design the calibration to compensate for both the non-linearity and non-reciprocity of BS transceivers. This allows alleviating the requirement to have dedicated hardware modules for transceiver linearization. Moreover, the proposed reciprocity calibration method is solely based on closed-form linear transformations, achieving a significant complexity reduction over state-of-the-art reciprocity methods, which assume linear transceivers, and rely on iterative methods. Simulation results showcase the potential of our approach in terms of the calibration matrix estimation error and downlink data-rates when applying zero-forcing (ZF) precoding after using our OTA-based DPD and reciprocity calibration method.