Having an egg limit per area could be a good idea to force players to buy new zones where production would be even better. But at the same time, having just one screen could be cool to keep it simple and give it a Tamagotchi-like vibe. It might even be a good idea to include a feature where you have to take care of the animals, like feeding them or giving them objects to play with.
Honestly, it depends a lot on the kind of atmosphere you want to create!