You can't put an Aura onto the battlefield this way unless that Aura can legally be attached to Evershrike. For example, you can't put an Aura card with "enchant player" onto the battlefield this way.

You can activate Evershrike's ability with X = 0. If you do, you'll return Evershrike to the battlefield from your graveyard, then exile it. (There are no Aura cards with mana value 0.)

You can activate Evershrike's ability only while it's in your graveyard.

You can activate Evershrike's ability even if you have no Auras in your hand, or if you have an Aura in your hand that you don't want to put on it or can't put on it. In either case, you'll return Evershrike to the battlefield from your graveyard, then exile it. No player has the opportunity to cast spells or activate abilities while Evershrike is on the battlefield.