In this paper, we investigate a secure cognitive radio network (CRN), which deploys non-orthogonal multiple access (NOMA) to deliver a mixed multicast and unicast traffic to the intended receivers, while keeping them secret from the eavesdroppers. This model represents a cognitive interference channel with an external eavesdropper (CIC-EE). In this model, there are one pair of primary nodes, one pair of secondary nodes, and an external eavesdropper. The primary transmitter multicasts a confidential message to both primary and secondary receivers, while trying to keep it secret from the eavesdropper. The secondary transmitter helps the primary user to deliver its message in exchange of transmission opportunity. The secondary message is unicasted to the secondary receiver, while concealing it from both primary receiver and the eavesdropper. This scenario models a NOMA-based overlay cognitive radio paradigm with an external eavesdropper. For this scenario, the achievable rate-equivocation region is obtained and its optimality is shown for a class of degraded channels. Then, the results obtained for the discrete memoryless channel are extended to the Gaussian channel model. Furthermore, by deploying numerical examples, a comparison is made between the proposed secure NOMA-based scheme, its orthogonal multiple access (OMA) based counterpart, and a cognitive interference channel without an external eavesdropper. It is shown that the NOMA-based method achieves significantly higher rates than its OMA based counterpart.