Both usages are correct, but the meanings are different. For example, I might say Australia boasts many fine beaches. This usage means Australia possesses fine beaches. If I brag about Australia and its beaches, then I am boasting of this.

"Boast" is intransitive, so you can only boast of something or boast about something (ie, you need prepositions). Although we know "boast" and "brag" well, the Australian word we often use is "skite".

I would probably say something like "He boasts that he's the best at football" or "He boasts about being the best at football", but I wouldn't be confused if you said one of the others. You could also use the word "brag" for a similar word. I hope this helps!