2nd Generation Eeveelution Methods
Unlike every other Pokemon in Pokemon GO, the method to obtaining a certain eevee evolution is at complete random. If a player were to evolve an Eevee without doing anything to that Eevee, the player would get either vaporeon, jolteon or flareon (not espeon or umbreon, even after the 2nd Generation Update). However, there is not one, but two ways for players to have a guarantee of evolving into either of the 2nd Generation Eeveelutions of Espeon and Umbreon.
The first way is very similar to the 1st Generation eeveelutions, revolving around giving Eevee a certain nickname to evolve it into a given form. If a player were to nickname their Eevee with the name of Sakura, it would evolve into Espeon; if a player were to nickname their Eevee with the name of Tamao, it would evolve into Umbreon. However, this method only works the first time you use it for Espeon and Umbreon. Every time this is done afterwards, even if given a certain nickname, Eevee will evolve into one of its Eeveelutions at random. (Again, Eevee can only evolve into Vaporeon, Jolteon or Flareon at random.)
The second method to getting espeon or umbreon is very similar to the method found in the Mainstream Pokemon Games. Within those games, eevee evolves into Espeon by leveling up during the day with maximum happiness and into Umbreon by leveling up during the night with the same criteria. In Pokemon GO, however, a player must have Eevee as their Buddy for ten kilometers and then evolve it. If they evolve it during the day after doing this, they will get Espeon; at night, and they will get Umbreon.