In low-power wireless neural recording tasks, signals must be compressed before transmission to extend battery life. Recently, Compressed Sensing (CS) theory has successfully demonstrated its potential in neural recording applications. In this study, a deep learning framework of quantized CS, termed BW-NQ-DNN, is proposed, which consists of a binary measurement matrix, a non-uniform quantizer, and a noniterative recovery solver. By training the BW-NQ-DNN, the three parts are jointly optimized. Experimental results on synthetic and real datasets reveal that BW-NQ-DNN not only drastically reduce the transmission bits but also outperforms the state-ofthe- art CS-based methods. On the challenging high compression ratio task, the proposed approach still achieves high recovery performance and spike classification accuracy. This framework is of great values to wireless neural recoding devices, and many variants can be straightforwardly derived for low-power wireless telemonitoring applications.