An Egglocke is a Nuzlocke variant where friends or subscribers send in
eggs to be hatched and used in the game instead of normal catches. The
normal Nuzlocke rules also apply, which are:
All Pokemon that faint have to be permanently boxed forever and be
You can only catch the first encounter you see in every location. Gift
Pokemon do not count towards this. If you already have something
that's is something that you have, you can skip the encounter and try
again. However, since it's an egglocke, the last part isn't a concern.
All your catches and gifted Pokemon have to be exchanged for a
received egg. Once the egg is hatched, use a Rare Candy cheat to level
the hatchling to the level of the Pokemon that you exchanged.
Those are the basic rules to any Nuzlocke or it's variant. They can be
changed or more rules can be added to what kind of challenge you want.
Via This post
So basically, normal Nuzlocke Rules, but with random eggs. You can only obtain one Pokemon per route, but you have to trade them off for a random egg. When the egg hatches, you level it up to use in your journey, and when the Pokemon faints, you release it. To start, pick your starter, and once you can, trade it in the chatroom (or elsewhere!) for a random egg.
I hope this helped!