Machine learning methods are increasingly applied to analyze health-related public discourse based on large-scale data, but questions remain regarding their ability to accurately detect different types of health sentiments. Especially, Large Language Models (LLMs) have gained attention as a powerful technology, yet their accuracy and feasibility in capturing different opinions and perspectives on health issues are largely unexplored. Thus, this research examines how accurate the three prominent LLMs (GPT, Gemini, and LLAMA) are in detecting risk-promoting versus health-supporting sentiments across two critical public health topics: Human Papillomavirus (HPV) vaccination and heated tobacco products (HTPs). Drawing on data from Facebook and Twitter, we curated multiple sets of messages supporting or opposing recommended health behaviors, supplemented with human annotations as the gold standard for sentiment classification. The findings indicate that all three LLMs generally demonstrate substantial accuracy in classifying risk-promoting and health-supporting sentiments, although notable discrepancies emerge by platform, health issue, and model type. Specifically, models often show higher accuracy for risk-promoting sentiment on Facebook, whereas health-supporting messages on Twitter are more accurately detected. An additional analysis also shows the challenges LLMs face in reliably detecting neutral messages. These results highlight the importance of carefully selecting and validating language models for public health analyses, particularly given potential biases in training data that may lead LLMs to overestimate or underestimate the prevalence of certain perspectives.