Emerging edge computing (EEC) has been introduced as an innovative paradigm for the healthcare applications of the Internet of Things (IoT) that aims to distribute the network resources at the network edges to improve security, communication, and decision-making processes. The operation of healthcare IoT applications typically needs the presence of interoperable modules. Despite numerous benefits, these applications face many security challenges at the network edge. In this context, advanced Artificial Intelligence techniques can be used at the network edges for these applications to efficiently utilize the available resources securely. To this end, we aim to present a detailed survey of healthcare IoT applications in the context of AI-enabled EEC technology to identify unresolved security challenges that need attention from the research community and healthcare stakeholders, and then suggest potential research directions to give a clear future insight.