The fight itself is no different than the normal Brumak fight. I employ shooters on opposite sides of the field and ping pong the creature them by alternating shots. If you've killed the Brumak before Jacked Mode debuted, your old strategy will work here, although probably slightly easier since Jack offers some control options. Cloaking is the only ability you really need for this on Jack, though others might help on the field.
For this acheevo, I kept Jack cloaked and out of the way, occasionally using his shock grenades to 'halt' grubholes or poison grenades to kill wretches. As long as you keep him out of missiles and rampage attacks, cloaked jack is not in any danger.
Edit: Jack had taken damage during the beginning of the stage before the boss fight and I still got the achievement, as did a commentator.