The Internet of Things (IoT) consists of resource-constrained smart devices capable to sense and process data. It connects a huge number of smart sensing devices, i.e., things, and heterogeneous networks. The IoT is incorporated into different applications, such as smart health, smart home, smart grid, etc. The concept of smart healthcare has emerged in different countries, where pilot projects of healthcare facilities are analyzed. In IoT-enabled healthcare systems, the security of IoT devices and associated data is very important, whereas Edge computing is a promising architecture that solves their computational and processing problems. Edge computing is economical and has the potential to provide low latency data services by improving the communication and computation speed of IoT devices in a healthcare system. In Edge-based IoT-enabled healthcare systems, load balancing, network optimization, and efficient resource utilization are accurately performed using artificial intelligence (AI), i.e., intelligent software-defined network (SDN) controller. SDN-based Edge computing is helpful in the efficient utilization of limited resources of IoT devices. However, these low powered devices and associated data (private sensitive data of patients) are prone to various security threats. Therefore, in this paper, we design a secure framework for SDN-based Edge computing in IoT-enabled healthcare system. In the proposed framework, the IoT devices are authenticated by the Edge servers using a lightweight authentication scheme. After authentication, these devices collect data from the patients and send them to the Edge servers for storage, processing, and analyses. The Edge servers are connected with an SDN controller, which performs load balancing, network optimization, and efficient resource utilization in the healthcare system. The proposed framework is evaluated using computer-based simulations. The results demonstrate that the proposed framework provides better solutions for IoT-enabled healthcare systems.