There are several synonyms for the phrase "keep your mouth shut." Some common alternatives include "zip your lips," "button your lip," and "stay quiet." Another option is to use the phrase "keep mum," which means to keep silent or keep a secret. You can also use "shush" to tell someone to be quiet or "hush up" to urge them to stop talking. If you want to convey the idea of secrecy, you could say "keep it under your hat" or "keep it to yourself." Lastly, if you want someone to stop talking or reveal too much information, you can simply say "enough said".