I think the eagles probably felt the same way about the ring as Tom Bombadil did. Gwaihir didn't mind lending a helping hand to Gandalf and his friends (namely Frodo and Sam) but asking him to risk life and limb for something that largely concerned the "free peoples of Middle Earth" would've been a bit much.
Aside from that, I seriously doubt that Gwaihir could've stood up to the Nine unless he had an army in which case the point of secrecy would be utterly defeated.
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"'You," he said, "tell her all. What good came to you? Do you rejoice that Maleldil became a man? Tell her of your joys, and of what profit you had when you made Maleldil and death acquainted.'" -Perelandra, by C.S. Lewis
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