The recent COVID-19 pandemic has brought significant challenges to the traditional healthcare industry. A novel eHealth system is required to cope with infectious and chronic diseases. This article proposes an online eHealth system to monitor patients and provides edge computing services based on the Internet of Unmanned Aerial Vehicles (UAVs). In order to minimize the health monitoring latency and guarantee the resource utilization efficiency of UAVs, the Lyapunov optimization method is utilized to decompose the long-term optimization problem into a series of instantaneous optimization problems. Then the decomposed sub-problem is formulated as a minimum cost maximum flow problem by constructing a bipartite graph that maps medical analysis requests to target UAVs. Extensive experimental results demonstrate the effectiveness of our system. Finally, system analyses illustrate that the robustness and scalability of our eHealth system is favorable, and our system can provide agile edge computing services for dispersed patients.