It is a pain in the Butt to get. They have to be "player-controlled", meaning a player has to be controlling them at the time you shoot them. So if they are running their AGR, Sentry, etc, you get credit. If the AI is running it, you are SOL.
The camouflages are unlocked via doing the following acts, unique to FHJ-18 AA:
¦DEVGRU - Destroy 5 aircraft.
¦A-TACS AU - Destroy 10 aircraft.
¦ERDL - Destroy 20 aircraft.
¦Siberia - Destroy 30 aircraft.
¦Choco - Destroy 40 aircraft.
¦Blue Tiger - Destroy 50 aircraft.
¦Bloodshot - Destroy 60 aircraft.
¦Ghostex: Delta 6 - Destroy 75 aircraft.
¦Kryptek: Typhon - Destroy 100 aircraft.
¦Carbon Fiber - Destroy 10 aircraft quickly as they enter the airspace.
¦Cherry Blossom - Destroy 5 aircraft in a single game.
¦Art of War - Destroy 2 aircraft back to back.
¦Ronin - Destroy 10 player-controlled scorestreaks.
¦Skulls - Destroy 1 A.G.R..
¦Gold - Complete all camouflage unlocks for this weapon.
¦Diamond - Have all launchers with gold camouflage.
¦Elite - Become an elite member.
I have yet to get my gold because of this challenge.
Ronin is a bit of a pain to unlock. The destroyed streaks that will definitely grant you progress include Dragonfires, VTOLs, and Lodestars. Others that have a small chance include Sentry Guns and AGRs (due to their user control abilities).
But look on the bright side--at least I have diamond camo.