Unequivocally not. Posting to Facebook is a public disclosure Health data: no one, including your own physician, is obligated to protect that which you opt to publicly disclose.
We are talking about a Facebook AI program reading your posts and deciding if you are suicidal or not. I'm not even sure how you would clinically validate that and if it would require FDA clearance, but if you are forming a mental health opinion on a stream of data and contacting the government about it that is probably something that may need to be protected.
The medical school structured itnas a clinical trial with people volunteering to participate. Medical studies have different regulations than just reviewing public postings.
This is a reflection of the way they’ve chosen to go about things - no doubt in response to past kerfuffles - rather than an actual legal obligation on fb’s analysis of public materials.
HIPAA regs aren’t hidden from the public. Peruse at your leisure.