The three attacking creatures don't have to be attacking the same player, planeswalker, or battle.

Once a battalion ability has triggered, it doesn't matter how many creatures are still attacking when that ability resolves.

Similarly, once you attack with three or more creatures in a single turn, you can still play the exiled card that turn no matter what happens to those creatures.

If Boros Strike-Captain leaves the battlefield, the effect allowing you to play the exiled card if you attacked with three or more creatures continues to apply, even during later turns, as long as you attack with three or more creatures during those later turns.

You pay all costs and follow all normal timing rules for a card played this way. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.