The rule of thumb here is that both are correct, in different contexts. You should use board certified when it comes after a verb, as in “He is board certified in cosmetic surgery,” but employ board-certified when you use it as an adjective before a noun, as in “He is a board-certified spine surgeon.”

Does board certified have a hyphen?

I found this explanation: Hyphenate when used as a compound adjective before a noun: "board-certified physicians." For other uses, drop the hyphen: "Dr. Smith is board certified in Pediatrics."

Do you capitalize physician specialties?

No, when used generically, medical specialities do not have to be capitalized. You may frequently see the specialities listed on a nameplate or office door where they are usually capitalized.