In this paper, output feedback control (OFC) stabilization of hidden Markov Boolean control networks (HMBCNs) is studied. Using semi-tensor product of matrices, the OFC problems to be solved are presented in algebraic form. All feasible OFC gains have been characterized. A special kind of attack on the HMBCNs, named shifting attack, is studied. Such kind of attack does not change the control policy via attacking the value of the output measurements, but via passing false information to the controller after receiving the output measurements. Necessary and sufficient conditions are presented for the security of the considered HMBCNs under shifting attacks. Besides, for those systems under attack, some constraints are considered. Necessary and sufficient conditions for the security of HMBCNs with $\gamma$-measurements protected are also presented. Moreover, the smallest number of measurements to be protected is further elaborated. Finally, a simulation example based on a biological system is used to illustrate the effectiveness of the obtained results.