In the first, it means to keep a secret. The person talking to you is asking or instructing you not to tell anyone else. Can you keep your lip zipped about it? It could be a secret, rumors, or gossip. Those are always good to zip your lip about, don't you think?
The second is just a request (or command) to keep quiet in general. If the person is extremely frustrated, they might use a shorter form. Zip It! ;)
An alternative that means the same thing is Button Your Lip (or Button It!).
Question, comments, suggestions ... please don't hesitate to let me know. Until next week! :)
