You get an Eevee egg from a guy in Esmerld city after you beat Team Steam. I don't remember who it is, but you should just talk to everyone. The guy that gives it to you will ask if your party is full when you talk to him, if you say no he'll give you the egg.
The only ways to get an Eevee is to either be given one or if you already have one or one of its evolutions (Jolteon, Flareon, Vaporeon, Espeon, Umbreon, Glacieon, or Leafeon) you can put that Pokemon and a Ditto in the daycare. They will then breed giving you an Eevee egg.