*you cannot get the boss achievements as a mercenary
Here is a guide for each of the boss battles. They can only be fought during weekend events. To fight the bosses you have to select the icon under mission select just like you would if you were playing an order mission. Each boss has territory requirements (the shield icons that are displayed under your team name). If you are not in a team with at least 2 "territory rating" you will need to join one. I highly recommend taking a full squad of 4 in the fight with you because you will need the firepower for these guys.
2 "territory rating"
This the easiest boss IMO. It starts out with 300,000 health and is weak to KE and CE. Gatling guns worked like a charm for my squad. Start out the battle by rushing and doing as much damage as possible before it throws up its shields. Keep your distance once it starts flying around because it will shoot giant lasers at everything underneath it. Its shields will still be up and you can't do any damage to it anyways. You should be able to destroy it when it lands before it is able to use its shields the second time.