What if you don't tell the eagles on the mission what's up? Even if corruptible, it would take at least some time for them to figure out what's going on and which of the hobbitses they are carrying has the ring.
The Eagles seem very picky about when they help and who they're helping. I highly doubt they would do such a deed without Gandalf letting them in on the full story.
Well, you can tell some Eagles that "there is a very important mission to go on with Gandalf, and we need some eagles who don't know why we're going on the mission."
Like, you could have the Army move a box of stuff for you, and tell the army, but still have none of the soldiers know what's in the box.
