The eagles are creation of vala manwe as the dwarves are the creation of vala aule. Maiar can only be created by eru, a vala cant create a maia but can use them as servants. As manwe creates the eagles they are not maiar.

@TThurst97, Tolkien changed his mind throughout the years on whether or not the Eagles were Maiar. Originally, where am I are spirits as shown in this quote from Morgoth's Ring: "Manwë however sent Maia spirits in Eagle form to dwell near Thangorodrim and keep watch on all that Melkor or did and assist the Noldor in extreme cases."

However later on he decided against it saying:

"The same sort of thing may be said of Húan and the Eagles: they were taught language by the Valar, and raised to a higher level– but they still had no fëar."

Hope this helps!

To answer your point about good old Tom Bombadil... Tom is Tom. He is not a Maia. He is not the manifestation of Eru. He is not any other idea (some of them silly) that have been imagined. He is only what is described in LOTR and The Adventures of Tom Bombadil and Other Verses from the Red Book, although the latter is just poetry. Tolkien was satisfied with him being a mystery and so we have to be satisfied with that too :)