The most common and most likely answer to this crossword clue is the 4 letter word HUSH
I've seen this crossword clue in Independent
H
U
S
H
Buddy explains!

The straight part
We know that "Shut up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case shut up = hush.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does HUSH mean?
HUSH noun- A silence, especially after some noise
- A mining method using water
- To become quiet.
- To make quiet.
- To appease; to allay; to soothe.
- To clear off soil and other materials overlying the bedrock.