I would hate to tell you this, anonymous, but very likely not, especially if your eevee feels so strongly about a certain evolution that he would resist doing so naturally, despite meeting all of the conditions for it. You could certainly try to outline his options in a tactful manner, but be sure to avoid comparing umbreon to flareon (as this may inadvertently cause him to feel that you would be disappointed in him if he chose not to evolve that way) and emphasize that you would support his evolution, regardless of which form he feels the most comfortable in. If anything, use this opportunity to test the strength of his convictions and ensure that he’s thought his decision all the way through, not because you wish to ensure he’s made the right choice with regards to his evolutionary form but instead because you wish to ensure he’s made the right choice in choosing to evolve at all.
This is because, ultimately, evolution is a permanent thing. While you’ve said you wouldn’t force your eevee to change his mind, this is something I feel is important for trainers to keep in mind generally speaking, as some trainers are not as conscious of their pokémon’s wishes as others. But when you get to the heart of the matter, evolution cannot be reversed, meaning the choice to evolve and the choice of which form to evolve into will affect a pokémon for the rest of their lives. That pokémon must be completely comfortable with both decisions before undergoing the process, or else the process itself will become a traumatic experience, either during or once the realization of what it means settles in. There have been cases of pokémon evolving to the “wrong” evolution (it happens most frequently with members of the oddish line for some reason), only to become far more aggressive once evolved.
So for the sake of your eevee’s sanity, I would suggest trusting his decision if he’s certain of it. If your eevee would be happiest being a flareon and you need an umbreon, there are certainly other options to take as a compromise, including allowing your eevee to evolve and then adopting a second eevee who would be far more open to the idea of filling that niche. Eevee and its evolutions are social pokémon when tamed anyway, so fitting a flareon and an umbreon on the same team can actually produce emotional benefits.