I've found that the green circles come up once the sails have been shot down so that the ship can't sail away. Switch from the cannon shot to the grappling hooks, and aim for the green circles. If your shot is successful, you'll see a rope slung between your ship and the enemy ship. The program will pull the ships closer. Exit your stance at the cannon and walk to the railing like you want to exit the ship. When the ship is close enough, the program will indicate "Press Control to board enemy ship" (or something similar).
Once you're on board, a variety of things could happen. Fight the soldiers. Take over the helm (although once a soldier got back to the helm after I left it, and I was stranded on this enemy ship, unable to exit the ship or take back over the helm with soldiers that kept respawning
). Or get hit with a bug. We crash enough times in the other parts of the program that it's worth the risk of a dc to see what will happen when you board the ship.
Happy hunting.