buttoned-up

(verb) refuse to talk or stop talking; fall silent. Synonyms: close up, clam up, dummy up, shut up, belt up, button up, be quiet, keep mum. Example: "The children shut up when their father approached."

(adj) (British colloquial) not inclined to conversation.

(adj) conservative in professional manner. Example: "Employers are looking for buttoned-up types."