Although reconfigurable intelligent surface (RIS) can improve the secrecy communication performance of wireless users, it still faces challenges such as limited coverage and double-fading effect. To address these issues, in this paper, we utilize a novel multi-functional RIS (MF-RIS) to enhance the secrecy performance of wireless users, and investigate the physical layer secrecy problem in non-orthogonal multiple access (NOMA) networks. Specifically, we derive the secrecy outage probability (SOP) and secrecy throughput expressions of users in MF-RIS-assisted NOMA networks with external and internal eavesdroppers. The asymptotic expressions for SOP and secrecy diversity order are also analyzed under high signal-to-noise ratio (SNR) conditions. Additionally, we examine the impact of receiver hardware limitations and error transmission-induced imperfect successive interference cancellation (SIC) on the secrecy performance. Numerical results indicate that: i) under the same power budget, the secrecy performance achieved by MF-RIS significantly outperforms active RIS and simultaneously transmitting and reflecting RIS; ii) with increasing power budget, residual interference caused by imperfect SIC surpasses thermal noise as the primary factor affecting secrecy capacity; and iii) deploying additional elements at the MF-RIS brings significant secrecy enhancements for the external eavesdropping scenario, in contrast to the internal eavesdropping case.