Better keep still about it

(someone had) better keep still about it

(Someone had) better not say anything about a particular issue or situation. "Someone" can be used in the phrase itself, especially when the listener already knows who the speaker is talking about. A: "I think Jake knows that we doctored the numbers in that report." B: "Well, someone had better keep still about it then." Better keep still about it so that the boss doesn't suspect anything.
See also: better, keep, still
Farlex Dictionary of Idioms.

Better keep still about it.

 and Someone had better keep still about it.; Better keep quiet about it.; Someone better keep quiet about it.
A particular person ought not to tell about or discuss something. (The someone can stand for any person's name, any pronoun, or even the word someone meaning "you-know-who." If there is no Someone had, the phrase is a mild admonition to keep quiet about something.) Mary: I saw you with Bill last night. Jane: You'd better keep quiet about it. Jane: Tom found out what you're giving Sally for her birthday. Bill: He had better keep quiet about it!
See also: better, keep, still
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.
See also:
  • (someone had) better keep quiet about it
  • (someone had) better keep still about it
  • (I'm) just saying
  • Will I see you again?
  • FWIW
  • IJS
  • 57
  • (one's) best foot forward
  • (as) sure as eggs (is eggs)
  • I don't suppose