No, I don't think wellbeing needs to be hyphenated.
It does though.
My spellchecker (or is that spell-checker?) accepts both spellings.
OK, you win!
Never mind, any spelling of "well-being" is better than "wellness" which OED calls "rather a nonce-wd" . Using it would mark you out as a Duolingo addict.
"Wellness " is infesting medical jargon recently.
What's wrong with "the health"?