Symbiotic radio (SR) networks are possible solutions to the future low-power wireless communications for massive Internet of Things devices. In this paper, we investigate a multiple-input-single-output non-orthogonal multiple access (NOMA) backscatter device (BD) aided SR network with a potential eavesdropper. In the network, a base station (BS) broadcasts signals to a central user and a cell-edge user using the NOMA protocol. With ambient backscatter modulation, the BD transmits its own messages to the central user over incident signals from the BS. We propose a constrained concave convex procedure-based algorithm which maximizes the $\epsilon$-outage secrecy rate from the BD to the central user under the achievable secrecy rate constraints from the BS to the central and cell-edge users. Simulation results illustrate that our proposed network achieves a much larger secrecy rate region than the orthogonal multiple access (OMA) network.