It's not an egg hunt in the sense that you find cleverly hidden eggs out in the wilderness. Instead you find extremely rare-spawning monsters called "Coddled Spriggans" and, if you're lucky enough to be the one to get the claim before all the other people camping the spawn point, you get the egg it drops.

Honestly, I would've preferred cleverly hidden eggs that randomly spawned around the world, using the same 'instancing' technique that they used for collecting items for quests and FATEs.