Eggs aren't supposed to do anything in Pokemon GO, other than eventually produce a new baby critter. Some players, however, are taking advantage of a new exploit that forces eggs to mess with the entire game.
Reports around the web are circulating that some players are finding unbeatable gyms. Somehow, players have figured out how to station eggs as gym leaders.
The thing is, eggs have no moves, and they have no HP. According to players, once an egg goes up on a gym, it cannot be defeated or dethroned. The egg just stays there and can continue to amass defender benefits to whatever jerk put it up. Players currently cannot do anything about egg gyms.
Right now, most of the images and footage floating around online are of borked gyms in NYC, where apparently some Team Instinct players have played eggs on gyms like the New York Times headquarters, among other locales:
I can't pinpoint any egg gyms around my area, though there are reports of this exploit spreading. We reached out to Niantic about the egg gyms, and will update this post if we hear back. For now it's worth noting that if you happen to find an egg gym, you can send Niantic a report here.
And if you're thinking of putting up an egg somewhere... don't. You'd be risking getting banned once the exploit is fixed.
Funny thing is, this isn't the first time eggs have wrecked havoc in a Pokémon game, as Bulbapedia explains below:
A Bad Egg (Japanese: ダメタマゴ Bad Egg), stylised as Bad EGG in Generation III, is a phenomenon present in the Generation III, Generation IV, Generation V, and Generation VI Pokémon games that results from a corruption of Pokémon data so that the checksum does not match up with the data's calculations. In Generation IV, sometimes Bad Eggs can hatch into — — - or another Bad Egg. Bad Eggs are not really Pokémon Eggs, and may possibly never have been, but instead the default message returned by the game if the checksum is wrong (which happens only if data is altered badly). Bad Eggs rarely hatch and cannot be released, merely taking up space, though there are methods of removing them — it is possible to trade a Bad Egg away into another game, or remove it by cheating.