I know what a Nuzlocke is but what of an Egglocke?
Also yes I could Google it but I might as well use this platform so someone else can benefit as well.
Like KRLW890 said you swap every Pokemon with an egg. Usually how this works is you get a ton of eggs and put them into your PC. Every time you catch a Pokemon you switch that Pokemon with an egg and hatch the egg. After hatching the egg, you would level it up it to the level of the Pokemon you swapped it with by using rare candies. To get the eggs you could ask friends to trade them to you, use a Pokemon ROM that has an egglocke feature, or ask people to gen the eggs in for you.
This video shows how to do an egglocke legit, albeit only in ORAS/XY
This video shows how to gen eggs, however I don't know how to send them into a game because I am unfamiliar with genning.
As far as I can tell, it seems pretty similar to a regular nuzlocke, except that "every caught Pokemon needs to be swapped out for an egg".