Six-Generation (6G) promises to revolutionize not only the way we understand wireless communications, but also all the interactions, services, and daily activities that are built upon it. The increase in device density and data rates combined with the reduction of energy needs and latency open the door to ground-breaking opportunities in multiple domains. In particular, smart health services could benefit from the adoption of 6G to become more efficient, resilient, sustainable, affordable, and ubiquitous. With the adoption of 6G, the current sensing, communication, and interaction capabilities of context-aware environments will be magnified, resulting in unprecedented, supercharged smart health services. However, despite all the advantages of 6G, health services must deal with sensitive data, and for 6G-based healthcare to be a reality, challenges related to security and privacy must be addressed from its inception. In this article, we present the opportunities and novel services that 6G technology enables for smart health. We provide a thorough discussion of the security aspects, by identifying specific security requirements, highlighting security challenges, and pointing out gaps that remain open and must be bridged by future research.