Is one allowed to eat hard "baked" eggs (not boiled in a pot) from a non kosher juice bar that only serves juices and eggs. Their oven is only used for eggs, as they don't serve any other food.
no, because there would be no way to verify such a claim in a non kosher kitchen. We don't rely on the word of non-Jews for koshrus.
If it is indeed only uses for eggs then there is no worry about it being not kosher in terms of ingredients. However it still falls under the category of bishul akum. So if you were to ask if it was okay for you to turn on their oven, or be a part of the cooking process in some way, then it would be permissible to eat.