There are quite a few Pokèballs Eevee can be in, but I'll try to name them all
Pokèballs Eevee can be in legitimately:
Pokèball: Catching Eevee in the wild, as a gift, breeding.
HA: Yes, through Dexnav and breeding
Great Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Ultra Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Master Ball: Catching Eevee in the wild. (But it is a waste of a Master Ball.)
HA: Yes, through Dexnav
Premier Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Repeat Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Dive Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Timer Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Nest Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Net Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Luxury Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Heal Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Quick Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Dusk Ball: Catching Eevee in the wild, breeding.
HA: Yes, through Dexnav and breeding
Cherish Ball: Events.
HA: No, not without hacks, there have never been any Eevee events for HA Eevee.
Pokèballs Eevee can be in illegitimately:
Level Ball
Lure Ball
Moon Ball
Love Ball
Friend Ball
Heavy Ball
Fast Ball
Reason: Eevee cannot be caught in the wild in Gold, Silver, Crystal, HeartGold, or SoulSilver.
Sport Ball
Reason: Eevee cannot be caught in the Bug Catching Contest in Gold, Silver, Crystal, HeartGold, or SoulSilver.
Safari Ball
Reason: Eevee cannot be caught in the Safari Zone in any game.
Hope this helps!
Source
Source (For Cherish Ball)
(Note: Cherish Balls and Master Balls cannot be passed down through breeding.)
