In this paper, we investigate the application of user cooperation (UC) and non-orthogonal multiple access (NOMA) schemes for a wireless powered mobile edge computing (MEC) system under the non-linear energy harvesting model, in which two single-antenna mobile users first harvest energy from a multi-antenna access point (AP) integrated with an MEC server. Then, during the computation offloading phase, both mobile users simultaneously offload tasks to the MEC server with the harvested energy, by performing NOMA protocol. To better enhance the system performance, UC scheme is carried out, where the near user acts as a relay to help the far user offload computation tasks to the AP. To obtain energy efficient MEC design, our objective is to maximize the computation efficiency (i.e., the total computation bits divided by the consumed energy) by jointly designing the energy beamforming, time and power allocations, which yields a challenging nonconvex optimization problem. To deal with it, the original problem is first transformed into a more tractable formulation by applying the semidefinite relaxation (SDR) technique and then solved by utilizing the sequential convex approximation (SCA) method. Numerical results demonstrate that UC has a great impact when two users are close, while NOMA makes effect when two users are relatively far. Combining both NOMA and UC, the proposed scheme, named NOMA-UC MEC, yields better system performance than the benchmark schemes.