An interesting topic... I think it would be very weird if the eagles wouldn't refuse to do that. They weren't very interested in the wars of men anyway, were they? And even if Gandalf would have one of the eagles consented to carry Frodo to Mount Doom, it would have been quite easy for Sauron or his servants to spot a great eagle with a hobbit on its back flying to Mordor, save flying straight to Mount Doom. Of course the eagle could have left Frodo somewhere outside Mordor, but how could Frodo then get over the mountains, for he would then be alone (for nine eagles carrying the fellowship would look a bit weird to Sauron, wouldn't it?) with not even Gollum helping him... If I'm right with the fact that it would be impossible for Gollum to follow an eagle flying high, if Gollum even knew about Frodo riding an eagle.
