They can't obviously do anything about the game itself. There is a Eggwars spigot plugin, I think. But don't quote me on that.
It should be possible to put a trademark on the title Eggwars, though.
While for the maps, considering they are works, more specifically art works, can not be used if not under proper license.
This would be, if our maps are copyrighted. I am not aware of that.
They aren't the only server with Eggwars. Some other servers have that title too. Again, with copied maps.
While it would be impossible to protect from plagiarism if they have alternate maps and an alternate name, even though the concept is the same, for example "BedWars" where you have to break the opponent's bed to prevent him from respawning.