Mobile edge computing (MEC) has been envisioned as a promising approach for enabling the computation-intensive yet latency-sensitive mobile Internet services in future wireless networks. In this paper, we investigate the secrecy based energy-efficient MEC via cooperative Non-orthogonal Multiple Access (NOMA) transmission. We consider that an edge-computing device (ED) offloads its computation-workload to the edge-computing server (ECS) subject to the overhearing-attack of a malicious eavesdropper. To enhance the secrecy of the ED's offloading transmission, a group of conventional wireless devices (WDs) are scheduled to form a NOMA-transmission group with the ED for sending data to the cellular base station (BS) while providing cooperative jamming to the eavesdropper. We formulate a joint optimization of the ED's offloaded workload, transmit-power, NOMA-transmission duration as well as the selection of the WDs, with the objective of minimizing the total energy consumption of the ED and the selected WDs, while subject to the ED's latency-requirement and the selected WDs' required data-volumes to deliver. Despite the nature of mixed binary and non-convex programming of the formulated problem, we exploit the vertical decomposition and propose a three-layered algorithm for solving it efficiently. To further address the fairness among different WDs, we investigate a system-wise utility maximization problem that accounts for the fairness in the WDs' delivered data and the total energy consumption of the ED and WDs. By exploiting our previously designed layered-algorithm, we further propose a stochastic learning based algorithm for determining each WD's optimal data-volume delivered. Numerical results are provided to validate the effectiveness of our proposed algorithms as well as the performance advantage of the secrecy based computation offloading via NOMA.