>>32573648I don't know much about competitive Pokemon, and Egg Moves in particular, but I believe Wish and Curse are the two big ones you'd want for your Eeveelutions. Vaporeon, Umbreon, and Sylveon in particular can benefit from one or both of them. But I don't think any other Egg Moves are known for being very useful for the Eeveelutions.
As for natures: I don't know exactly what natures are best for which Eeveelution, other than Timid is good for Espeon and Jolteon. Smogon's entries on the Eeveelutions could point you in the right direction, but be warned that they pick natures specifically to fit their recommended sets. So if you decide to use a different set than what you see on Smogon, a different nature might be better.
