The most common and most likely answer to this crossword clue is the 4 letter word CROW
I've seen this crossword clue in The Times
C
R
O
W
More clues leading to the result CROW
Buddy explains!

The straight part
We know that "Boast about" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case boast about = crow.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 CROW mean?
CROW noun- A bird, usually black, of the genus Corvus, having a strong conical beak, with projecting bristles; it has a harsh, croaking call.
- The cry of the rooster.
- Any of various dark-coloured nymphalid butterflies of the genus Euploea.
- A bar of iron with a beak, crook, or claw; a bar of iron used as a lever; a crowbar.
- A gangplank (corvus) used by the Ancient Roman navy to board enemy ships.
- (among butchers) The mesentery of an animal.
- A black person.
- The emblem of an eagle, a sign of military rank.
- To make the shrill sound characteristic of a rooster; to make a sound in this manner, either in gaiety, joy, pleasure, or defiance.
- To shout in exultation or defiance; to brag.
- To test the reed of a double reed instrument by placing the reed alone in the mouth and blowing it.