Unlike the previous two games were the day care man would call you over if he had a egg for you, the caretaker in X and Y doesn't call your attention to the fact. You have to check, and he may or may not have anything for you. However, there's a way to tell without heading over to engage him in conversation. If he faces the road, that means he has an egg. Otherwise, he'll be looking toward the building. Knowing that fact can save you some hassle.