No, they couldn't have given it to the Eagles. Many reasons.
1) There would be no story
2) The Nazgul and the Eye of Sauron would easily see a gigantic bird flying towards Mount Doom.
3) How do you expect them to carry it? Their huge talon? It would easily slip out into hiding yet again!
4) Gwaihir himself told Gandalf that he could only bear him a short distance from Isengard. The Eagle's business is not with the people of Middle-earth.
Also, Tolkien writes on why the Eagles would not have taken the Ring into Mordor. It might be in that link the last poster gave, I'm not sure.
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