button表示：用纽扣扣紧；button your lip字面意思是像扣子一样扣紧你的嘴唇，实际就是：闭嘴；住嘴；闭口不谈。

【例句】
Our boss plans to promote Bob, who is neither responsible nor capable. But you'd better button your lip and not say a word against it, because Bob is nobody but the boss's brother-in-law.
老板打算提升Bob这个既不负责责任又不能干的人，然而你最好还是闭口不提反对意见，因为Bob是老板的内弟。

Her boss ordered her to zip my lip.
她的老板命令她住口。