Memristor characteristics such as nonlinear dynamics, state retention and accumulation are useful for many applications. FPGA implementation of memristor-based systems and algorithms provides fast development and verification platform. In this work, we first propose a versatile digital memristor emulator that exhibits either continuous or discrete behaviors, similar to valence change memories (VCM) and the electrochemical metallization memories. Secondly, the proposed memristor emulator is used to design a chaotic generator circuit utilizing the memristor's nonlinearity. Finally, the chaotic system is used to design a speech encryption engine to demonstrate its capabilities. The memristor emulator, chaotic generator, and the encryption system were implemented on Nexys 4 Artix-7 FPGA XC7A100T. The implementation results show an efficiency in throughput and hardware resources utilization compared to the previous works. In addition, the encryption system results show good performance against several perceptual, statistical attacks in addition to resistance to security attacks tests including differential attacks, NIST tests, key space analysis, mean square error (MSE), correlation, histogram and spectrogram.