The eagles could be corrupted, yeah. They're sentient and (some at least) are ancient. One of the central themes of The Lord of the Rings is that the Powers of Middle Earth are very, very tempted by the ring and must overcome the test of the ring at cost to themselves: Galadriel is doomed to travel to the West, Boromir loses his mind temporarily, Saruman becomes a tyrant and eventually his staff broken etc. The eagles are themselves a Power and if they picked a Ring Bearer up they'd be tempted to drop said bearer on the ground and fetch the ring from their dead hand, become some type of eagle tyrant.

Maybe they would, maybe they wouldn't. Big risk either way.

Anyway, this theme is why the most meek people of Middle Earth -- the Hobbits, who desire basically nothing more than a quiet life -- are the only ones in Middle Earth that could actually cart the ring into Mordor. Even then it's not a sure bet, considering Gollum.