A promising effective human-robot interaction in assistive robotic systems is gaze-based control. However, current gaze-based assistive systems mainly help users with basic grasping actions, offering limited support. Moreover, the restricted intent recognition capability constrains the assistive system's ability to provide diverse assistance functions. In this paper, we propose an open implicit intention recognition framework powered by Large Language Model (LLM) and Vision Foundation Model (VFM), which can process gaze input and recognize user intents that are not confined to predefined or specific scenarios. Furthermore, we implement a gaze-driven LLM-enhanced assistive robot system (MindEye-OmniAssist) that recognizes user's intentions through gaze and assists in completing task. To achieve this, the system utilizes open vocabulary object detector, intention recognition network and LLM to infer their full intentions. By integrating eye movement feedback and LLM, it generates action sequences to assist the user in completing tasks. Real-world experiments have been conducted for assistive tasks, and the system achieved an overall success rate of 41/55 across various undefined tasks. Preliminary results show that the proposed method holds the potential to provide a more user-friendly human-computer interaction interface and significantly enhance the versatility and effectiveness of assistive systems by supporting more complex and diverse task.