Possible Clues:

• Is tight-lipped
• Goes mum
• Goes quiet
• Zips one's lip
• Is mum's seafood chowder out of the ordinary?
• Figuratively, stops talking ... or, literally, what are hidden in this puzzle's four longest answers
• Goes mum, and hint to what 3-, 9-, 10-, and 14-Down literally have in common
• Is mum making chowder with nothing missing?
• Stops talking about bashes at Oxford?
• Stops talking