We investigate the problem of secure non-orthogonal multiple access (NOMA) against full-duplex proactive eavesdropping, where the eavesdropper performs passive eavesdropping and active jamming simultaneously to interrupt the NOMA transmissions. To avoid the transmission outage caused by the unknown jamming level from the eavesdropper, we propose a novel transmission outage constrained scheme to limit the transmission outage probabilities of the users to a maximum tolerable threshold, which is also helpful in reducing the secrecy outage. We derive analytical expressions for the secrecy outage probability and secrecy diversity order to characterize the secrecy performance. Simulation results are provided to demonstrate the accuracy of the derived analytical results and the efficiency of the proposed scheme.