In future 6G networks, mobile edge computing (MEC) is envisioned to offer integrated computing, communication, and storage services at the network edge, enhancing both computational efficiency and communication quality. However, most existing MEC designs neglect the impact of jamming attacks, especially those from intelligent and adaptive jammers. To fill this important research gap, this paper investigates a jamming-resilient MEC framework that aims to improve communication reliability and reduce system delay under adversarial interference. Leveraging the emerging movable antenna (MA) technology, which allows dynamic adjustment of antenna positions and orientations, we propose a novel MA-assisted anti-jamming MEC architecture. Unlike existing works, our model explicitly considers the movement delay caused by MA, which is critical for practical deployment. We jointly optimize the MA positions at both the user equipment (UE) and the base station (BS), BS transmit beamforming, and task offloading ratios to minimize the total system delay. The resulting optimization problem is non-convex and highly coupled. Thus, we develop an efficient algorithm based on penalty dual decomposition (PDD) and successive convex approximation (SCA). Simulation results demonstrate that the proposed scheme significantly outperforms traditional fixed-position antenna (FPA) baselines in terms of jamming resilience and delay minimization, offering new insights into robust MEC system design for 6G networks.