Remote and resource-constrained Internet-of-Things (IoT) deployments often lack terrestrial connectivity for task offloading, motivating non-terrestrial networks (NTNs) with onboard multiaccess edge computing (MEC) capabilities. Nevertheless, in the presence of malicious actors, authentication needs to be performed to avoid non-authorized nodes from draining the computing resources of the NTN nodes. As a solution, we propose a four-layer MEC-enabled NTN with unmanned aerial vehicles (UAVs) acting as access nodes, a high altitude platform station (HAPS) acting as coordinator and authenticator, and a constellation of low-Earth orbit satellites (LEOSats) acting as remote MEC servers. We consider a tag-based physical-layer authentication (PLA) scheme to authenticate legitimate users, and formulate a joint task offloading decision and resource allocation for the admitted tasks, which is solved via block coordinate descent. Numerical results show that the PLA scheme is efficient and performs better than the benchmark schemes. We also demonstrate that the proposed scheme is robust against malicious attacks even under relaxed false-alarm constraints.