Back when this game was new, this was a fairly easy achievement to get, but now that nobody plays the multiplayer anymore, it's a bit more tricky, but there are pretty easy ways around it.
So here's the objective for this achievement (credit to Maka91): "For this achievement you must shoot all 4 hive locations located in the 4 new multiplayer maps that are a part of the Devastation DLC. You can quit the game once you shoot the hive and the notice pops up.
These MUST be done in public matches."
Ok, so the chances of getting a DLC map in the matchmaking system is extremely rare nowadays, and not worth wasting time over. The way you want to do this achievement is to either have a buddy help out or use a secondary Steam account you can use Family game sharing with.
In CoD: Ghosts Multiplayer, there are 3 main modes, Multiplayer, Extinction, and Squads. Most people believe that you have to play Multiplayer to get this achievement since it doesn't work in private matches but that is actually not true. Every "Egg-Stra XP" achievement WILL work in Squads which can be played solo. To get this achievement in Squads, use any of the methods below.
Method 1: Playing Wargame.
One of the gamemodes in Squads is called Wargame, this will set you against a random Squad on a random map on a random gamemode. This method might take longer or faster depending on your luck, since it picks random maps from your total map pool. So basically for this method, you load into Squads and choose Wargame, you can pick either difficulty, Recruit will be easier. Next, ignore the Find Game button, and instead click Play Now at the bottom. Now it will select a random map out of your pool. If it doesn't give you a map from this DLC, then go back, and select Play Now again, continue doing this until you get a map from the Devastation DLC pack. Now watch the video guide and shoot where you need to. You can leave the match once it registers. Now rinse and repeat for all the other maps in this DLC pack. Once you have gotten them all, then go into Extinction and play the map Mayday. Now destroy the first hive and the achievement should pop.
Method 2: Using a friend.
For this method, you need a buddy who owns at least the Devastation DLC, but preferably the entire season pass so they can help you get the other Egg-stra XP achievements. Get your friend to go into Squads mode, and change their Squad HQ to any of the four Devastation DLC maps. Once they have it set, then load into Squads, then choose Squad Assault, then at the bottom you will see Challenge Friend. Select this and choose your buddy who is helping you out. This should throw you into a game set to their Squad HQ which should be the DLC map you had them pick (If it isn't then restart the game and try again). Now watch the video locations guide, and shoot the Hive. Now once you're done have your friend switch to the second Devastation DLC map, rinse and repeat until you have shot each hive in all four maps. Once you have gotten them all, then go into Extinction and play the map Mayday. Now destroy the first hive and the achievement should pop. Now thank your buddy and help them get the achievement if they don't have it.
Method 3: Using an extra Steam account.
If you don't have any friends who play Ghosts then don't worry, all you need is an extra Steam account. If you don't have an extra account, then go make one. Once you have the account set up you can enable Family game sharing on your primary account, this will allow you to share your copy of Ghosts (along with all DLC) with your second account. Once you get that set up, now you need to log into the second account and start Ghosts Multiplayer. If you have a second PC that can load into the Ghosts menus then that will make this much easier, if not, it will be more tedious, but still doable on one PC. Once in the game on your second account, go to Squads, then Squad HQ, select Edit HQ Base, and change it to any of the four Devastation DLC maps. Now go back to your main account, and load up the Squad Assault gamemode, and select challenge friend at the bottom. Now challenge your second account and it should put you into the DLC map that you picked (if not then restart the game and try again). Now once you've shot the hive, have the second account switch to the next Devastation DLC map. Rinse and repeat until you have shot each hive across the four Devastation DLC maps. Once you have gotten them all, then go into Extinction and play the map Mayday. Now destroy the first hive and the achievement should pop.
Devastation DLC Maps:
Ruins
Behemoth
Collision
Unearthed
Mayday (Extinction)
Check out my guides for the other three Egg-stra XP achievements: