Holy crap Eevee is a bitch to catch or what? I just wasted 6 Pokeballs on it. I had him paralyzed and with only 1 HP left and it still wouldn't get caught. Eventually died because of Leech Seed :/.
Even Snorlax was much easier.
I had to use a couple of ultraballs to catch him...and just one of 'em. Lol