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Well since everyone else answered the question I feel left out and will repeat the answer also, but will try to summarize all the answers.
You CAN repeat the Boss Battle after having completed it on your own story quest.
You can do this by being crewed up/invited for another pirates Boss Battle, or
Using Lookout (L) you can go to Treasure Maps, and initiate the Boss Battle there.
You can take whatever crew (except Basic members) with you on the Boss Battle.
You cannot "solo" the Boss Battle, if you attempt to do it without a crew, it will place you in someone elses Boss Battle.
You could "solo" the Boss Battle (with a little work around) by crewing a friend or two, and either having them log out upon first phase of the Boss Battle, or by asking them not to intervene (just spectate).
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