People are increasingly seeking healthcare information from large language models (LLMs) via interactive chatbots, yet the nature and inherent risks of these conversations remain largely unexplored. In this paper, we filter large-scale conversational AI datasets to achieve HealthChat-11K, a curated dataset of 11K real-world conversations composed of 25K user messages. We use HealthChat-11K and a clinician-driven taxonomy for how users interact with LLMs when seeking healthcare information in order to systematically study user interactions across 21 distinct health specialties. Our analysis reveals insights into the nature of how and why users seek health information, such as common interactions, instances of incomplete context, affective behaviors, and interactions (e.g., leading questions) that can induce sycophancy, underscoring the need for improvements in the healthcare support capabilities of LLMs deployed as conversational AI. Code and artifacts to retrieve our analyses and combine them into a curated dataset can be found here: https://github.com/yahskapar/HealthChat