In future 6G networks, anti-jamming will become a critical challenge, particularly with the development of intelligent jammers that can initiate malicious interference, posing a significant security threat to communication transmission. Additionally, 6G networks have introduced mobile edge computing (MEC) technology to reduce system delay for edge user equipment (UEs). Thus, one of the key challenges in wireless communications is minimizing the system delay while mitigating interference and improving the communication rate. However, the current fixed-position antenna (FPA) techniques have limited degrees of freedom (DoF) and high power consumption, making them inadequate for communication in highly interfering environments. To address these challenges, this paper proposes a novel MEC anti-jamming communication architecture supported by mobile antenna (MA) technology. The core of the MA technique lies in optimizing the position of the antennas to increase DoF. The increase in DoF enhances the system's anti-jamming capabilities and reduces system delay. In this study, our goal is to reduce system delay while ensuring communication security and computational requirements. We design the position of MAs for UEs and the base station (BS), optimize the transmit beamforming at the UEs and the receive beamforming at the BS, and adjust the offloading rates and resource allocation for computation tasks at the MEC server. Since the optimization problem is a non-convex multi-variable coupled problem, we propose an algorithm based on penalty dual decomposition (PDD) combined with successive convex approximation (SCA). The simulation results demonstrate that the proposed MA architecture and the corresponding schemes offer superior anti-jamming capabilities and reduce the system delay compared to FPA.