I don't think the eagles were maiar. (I have nothing to prove that, so maybe I shouldn't be saying that.) Or maybe Thorondor was, and so those who descended from him had a bit of maia-blood in their veins. Or probably not, because when speaking about the mightiest of maiar, Thorondor is never mentioned. So probably he is not a maia.
Gurthang, you have a good question. I don't think that the eagle was spying for Sauron; eagles are clearly portrayed to be against him. I don't either think that the eagle was to warn the fellowship; how would just flying over them warn them? (Well, Legolas and Aragorn got worried but...)
It just came to my mind that Gandalf mentions the eagle and says it was Gwaihir.
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Like the stars chase the sun, over the glowing hill I will conquer
Blood is running deep, some things never sleep Double Fenris
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