Train them, actually. Not training in the battling sense, of course, but rather training in the positive reinforcement sense. A few people have encountered similar issues, so this link should offer several tips: http://bills-pokedex.tumblr.com/search/separation
The key is to be patient and start with a schedule. Schedules are often comforting to pokémon because they’re a constant. A pokémon understands time a little better once they’re on a schedule, and consequently, it’s easier for them to feel normal and safe. Thus, once you establish set times for when your eevee needs to be fed, walked, and so forth, it will be easier to change your routine to include independence.
Or in other words, once you set her on a schedule, then you can train her to handle herself on her own. For example, announce when you’ll leave her side, even for a moment, when you set her down to eat. Get her used to that idea, then use the same cue to announce when you need to leave the room, to leave the house temporarily, to leave for work, and so on and so forth. Start with small increments and work your way up so she grows more and more used to this routine.
Additionally, reward her for behavior in which she handles herself without reacting badly. If, for example, she doesn’t immediately tear off looking for you when you set her down and leave the room, give her a treat or a toy. If she does tear off, sit her back down and try again. Additionally, give her a treat before you have to leave her alone for long periods of time (for example, to take a shower at first, then work your way up to leaving the house) so she associates your absence with something good. Finally, give her something that smells like you to comfort her in between. The more she understands that not clinging to you (literally) is okay, the easier it will be to wean herself off you, as it were.
Good luck!