The most common and most likely answer to this crossword clue is the 4 letter word HIST
H
I
S
T
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 = hist.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 HIST mean?
HIST noun- An instance of an exclamation attracting attention or injunction to be silent.
- An utterance used to discreetly attract someone's attention.
- An injunction to be silent and/or to pay attention to what is being said or can be heard.
- The aggregate of past events.
- The branch of knowledge that studies the past; the assessment of notable events.
- A set of events involving an entity.
- A record or narrative description of past events.
- A list of past and continuing medical conditions of an individual or family.
- A record of previous user events, especially of visited web pages in a browser.
- Something that no longer exists or is no longer relevant.
- Shared experience or interaction.
- To raise; to lift; to elevate (especially, to raise or lift to a desired elevation, by means of tackle or pulley, said of a sail, a flag, a heavy package or weight).
- To lift a trophy or similar prize into the air in celebration of a victory.
- To lift someone up to be flogged.
- To be lifted up.
- To extract (code) from a loop construct as part of optimization.
- To steal, to rob.