Silmarillion chapter 2, Of Aule and Yavanna, states that the Eagles were neither Valar nor Maiar but instead offspring of the thoughts of Manwe and Yavanna, given life by Iluvatar:
But dost them not now remember, Kementari, that thy thought sang not always alone? Did not thy thought and mine meet also, so that we took wing together like great birds that soar above the clouds? That also shall come to be by the heed of Iluvatar, and before the Children awake there shall go forth with wings like the wind the Eagles of the Lords of the West.
This was largely derived from an essay called Of the Ents and Eagles, published in HoME11, and which Christopher Tolkien notes was used largely unmodified in the Silmarillion.
It's frequently speculated that many other types of spirit or being in Tolkien's works where doubt exists about their origin must be Maiar. This IMO is a very restrictive viewpoint.
The Valar and Maiar have a very specific origin: they're members of the Ainur, spirits that were created before the world, and that took part in the Music. They came down into the world following it's creation, and live in it but are not of it. The Ainulindale is clear about this:
...their shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment...
The Valaquenta is also clear about the distinction between Valar and Maiar only being in degree of power, but they're otherwise of the same class of being:
With the Valar came other spirits whose being also began before the World, of the same order as the Valar but of less degree. These are the Maiar, the people of the Valar, and their servants and helpers.
Eagles are an example of another class of spirit, one that was sent to the world by Iluvatar after it's creation, or that was created as part of the world. We see similar in the extract from Of Aule and Yavanna/Of the Ents and Eagles preceding that I give above:
Behold! When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar, and some will dwell therein...
These kind of spirits are also mentioned in the Ainulindale, and their origin was that they were called by Manwe after the Valar and Maiar had entered the world and to aid in their first conflict with Melkor:
...he called unto himself many spirits both greater and less, and they came down into the fields of Arda and aided Manwe, lest Melkor should hinder the fulfilment of their labour for
So the inescapable conclusion is that as well as the Valar and Maiar, there are other spirits in the world, and these were the source of the Eagles, the Ents, and yes, probably even Bombadil and Goldberry.
Update - 24th December 2014
The Eagles were actually probably intended to be Maiar.
This derives from a note in the Annals of Aman (History of Middle-earth 10) where it's said:
Manwe however sent Maia spirits in Eagle form to dwell near Thangorodrim and keep watch on all that Melkor did...
I'm uncertain at the moment if this was Tolkien's last words on the subject, or if Of the Ents and Eagles supersedes it.
Update - 28th March 2015
I had forgotten this. Reviewing Of the Ents and the Eagles, Christopher Tolkien notes:
This brief text belongs to the late, or last, period of my father's work, and must be dated at the earliest to 1958-9, but may well be later than that.
The note which I referred to above was written on the typescript of the Annals of Aman, which Christopher Tolkien dates to 1958, so while the relation of the two texts is not definitively confirmed, it seems that Of the Ents and the Eagles is probably the later work, and intended to stand as an authorative origin-story for both.