With the development of artificial intelligence integrated next-generation communication networks, mobile users (MUs) are increasingly demanding the efficient processing of computation-intensive and latency-sensitive tasks. However, existing mobile computing networks struggle to support the rapidly growing computational needs of the MUs. Fortunately, space-air-ground integrated network (SAGIN) supported mobile edge computing (MEC) is regarded as an effective solution, offering the MUs multi-tier and efficient computing services. In this paper, we consider an SAGIN supported MEC system, where a low Earth orbit satellite and multiple unmanned aerial vehicles (UAVs) are dispatched to provide computing services for MUs. An energy efficiency maximization problem is formulated, with the joint optimization of the MU-UAV association, the UAV trajectory, the task offloading decision, the computing frequency, and the transmission power control. Since the problem is non-convex, we decompose it into four subproblems, and propose an alternating optimization based algorithm to solve it. Simulation results confirm that the proposed algorithm outperforms the benchmarks.