I can't stress enough how much Espeon does not have to be a special sweeper just because of its high Special Attack and Speed stats. Assuming this does a great disservice to the sexiness that is Calm Mind and Baton Pass, and Espeon can arguably pull this combination off the best, being only weak to Dark attacks specially. I honestly think that nothing can really "counter" EspyJump straight up, with Baton Pass and great Speed, besides Dugtrio if you don't have a Substitute up and are at full or near-full HP (Adamant Dugtrio does a maximum of 79% damage with Earthquake).
Baton Pass is your friend. So is Calm Mind. You are encouraged to have a good special recipient of EspyJump's Calm Minds, though the Special Defense boost is always welcomed. You can bring it out against a non-STAB-boosted special attack as well, and then use your head and try to outpredict your opponent's status or attacking attempts. And honestly, I'm trying to think of a better Celebi counter than EspyJump, and I really can't. If you manage to status some of your opponent's Pokémon and Celebi pops out, you should immediately be salivating at your keyboard. The predictable Heal Bell, and the high probability that Espeon's "nemesis" Blissey is not also being used by your opponent, gives you a great opportunity to set EspyJump up if you play it smart, and Substitute regardless of whether they Leech Seed or switch.
Hidden Power Fire and Hidden Power Water can go over Baton Pass if you want a chance to take down some common Pokémon that think they've got Espeon in the bag: Metagross, Tyranitar, and Houndoom. You can also use Morning Sun for more staying power. Bite is an option for other Psychic-types but I wouldn't recommend it, because the more powerful Hidden Power Dark would be better since the Pokémon that are weak to Dark attacks, Psychics and Ghosts, are all usually faster than Espeon (read: Alakazam, Gengar, Starmie), so you won't be flinching the Pokémon that Bite is super effective against. Charm is there for annoyance, although it is still a move better suited for a Pokémon that is designed to actually take physical blows, like Umbreon. The same goes for Wish, really. Reflect can let Espeon keep up a Substitute against both physical and special attacks, but the only thing you could really drop for that is Psychic, which is just going to make life easier for phazers like Skarmory.
EspyJump, surprising as it may seem, is a special tank. It is there to lull special attackers into a false sense of security, and then belie its charming exterior by setting up until your opponent realizes you've got three Calm Minds and a Substitute and there's no way that they can break your Substitutes efficiently. For this reason, I prescribe a Timid nature, putting 200 EVs in HP for 321, 236 in SpD for 285 and 72 in Speed for 301. This used to be 296 Speed because when I devised EspyJump in 2004, Choice Band Salamence and its 299 Speed had yet to be created; rather, Heracross and its max-295 speed was the thing to beat. 321 HP is used over 320, as this allows for an extra Substitute, since it's conceivable that Espeon takes no direct damage, thanks to Calm Mind and Substitute blocking attacks, especially if someone attempts to status Espeon, giving you a free turn. Remember, Espeon is not Alakazam, and it doesn't need to outspeed that much, especially if you have a Substitute up and can Baton Pass your Calm Minds to an appropriate teammate anyway before threats like Tauros and Slaking can get you.
Checks and Counters
The Aerodactyl sacrifice and Dugtrio combo can counter this and like 90% of the things in the game, but I'll mention it again. The aforementioned Metagross, Tyranitar, and Houndoom are all effective counters to EspyJump somewhat, as is Claydol (they will all fall to Hidden Power Fire/Water though), but that is what Baton Pass is for, being faster than all of those Pokémon save Houndoom, whose attack you should be protected against via Substitute or Calm Mind anyway. Tyranitar messes with EspyJump the most, its Sand Stream negating the Leftovers that are integral to its efficacy. And of course I can't forget the obligatory Blissey mention, but again, Baton Pass (to a CursePert for example) makes the strategy behind EspyJump virtually uncounterable. Snorlax also forces a Baton Pass, and Skarmory and Suicune will Roar you the hell out if you don't attack them soon enough, and they of course aren't OHKOed anyway. Again, I feel that nothing can really be a "counter" to EspyJump unless it manages to phaze it, which Skarmory and a Roaring Suicune can do, which is why you shouldn't always just sit there and set up. If you're worried about getting Roared out, jack up its Special Attack—I'm an extremist and I know that putting absolutely no EVs into a Special Attack is so sexy (394 highest) is somewhat of a crime.
Jumpman16's Final Thoughts
Oh EspyJump, you are my first love. This Pokémon has no business being Underused still, and I feel that it is underused because most players think that it is just a weaker version of Alakazam. Espeon was just about my favorite Pokémon in GS because of its looks, and with what I learned it can do in Advance play my love for it just skyrocketed. It is an awesome Pokémon, and I suggest you all give it a try before thinking it should still stay on the UU shelf.