The Eye would have saw them coming. The whole point of the mission was to do it by stealth and have Sauron not suspect a thing as the idea was so implausible - flying big eagles into his territory would have been a bit of a dead giveaway.
2. He has the Nazgul in the air to counter the eagles. And countless archers and other projectile throwers. Even if they tried one eagle carrying the ring in a pouch and tried to drop it right in the volcano, the risk would have been obscene; if it's shot down Sauron has the ring and it's game, set and match.