To remain silent, especially in order to keep a secret or to avoid saying something inappropriate.
To fasten with a button or buttons.
Button up your shirt cuff. It has come undone
To fasten all the buttons on a coat, or similar item of clothing, to keep warm.
Button up well. It's freezing outside today.
To conclude; to finalize; to put the finishing touches (on)
It's time to button up the meeting.
To seal off (a military vehicle or facility).