In this paper, we study a two-phase underlay cognitive relay network, where there exists an eavesdropper who can overhear the message. The secure data transmission from the secondary source to secondary destination is assisted by two decode-and-forward (DF) relays. Although the traditional opportunistic relaying technique can choose one relay to provide the best secure performance, it needs to continuously have the channel state information (CSI) of both relays, and may result in a high relay switching rate. To overcome these limitations, a secure switch-and-stay combining (SSSC) protocol is proposed where only one out of the two relays is activated to assist the secure data transmission, and the secure relay switching occurs when the relay cannot support the secure communication any longer. This security switching is assisted by either instantaneous or statistical eavesdropping CSI. For these two cases, we study the system secure performance of SSSC protocol, by deriving the analytical secrecy outage probability as well as an asymptotic expression for the high main-to-eavesdropper ratio (MER) region. We show that SSSC can substantially reduce the system complexity while achieving or approaching the full diversity order of opportunistic relaying in the presence of the instantaneous or statistical eavesdropping CSI.