As exciting as that might be, anonymous, unfortunately no. Evolution is a full-on transformation into a set form: there is no such thing as a hybrid evolution (in this sense, anyway—some people argue that slowbro and slowking are hybrid evolutions, but that’s another story) for the same reason hybrid hatchlings don’t exist. They’re two separate genetic entities, in other words, defined by very specific criteria. Think of it like a light switch, in other words, wherein the conditions for the state of being “on” are completely different from those of being “off.”
That having been said, an eevee in this scenario would evolve based on which emotion is stronger. If they feel generally happier, they’ll evolve into either espeon or umbreon, depending on the time of day. (Eevee will evolve into espeon if they’re exposed to any form of sunlight, including the little amount they get at sunrise and sunset, incidentally.) If they feel a stronger sense of belonging and physical comfort, they’ll be sylveon instead. Usually, one emotion is stronger than the other (either happiness for their trainer or pack or the feeling of being loved by their trainer or pack), but if the feeling is exactly equal, they’ll remain as eevee until one emotion outweighs the other.
Unless, of course, the eevee has chosen its particular form. If an eevee at equilibrium actually wants to be an umbreon instead of a sylveon or vice-versa, then they’ll will themselves towards that form.
This is, of course, also not taking into consideration environmental factors (such as proximity to a moss-covered rock), but seeing as you’ve specified emotion-based evolutions for eevee, I’ll keep it at this.