*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.
Air Master- Raijin
1 "territory rating"
This one is pretty easy. Raijin will start by flying around the canyon. Take it down as soon as you can (Someone in my squad shot it down with a sniper cannon very easily, if you are having trouble taking it down try this). Once it crash lands it will spawn a laser cannon which you will want to keep your distance from!! Just follow the mission path and take your time to kill all of the turrets along the way. When you get to the landing site take the right path (with 10+ turrets) because you will be out of the line of fire from the giant laser. Once you clear it out use CE (KE works very well also) and destroy the cannon which will end the mission.