Argh! The 'Eagle' question!
Honestly, this one has been tossed around so often where I work that it's now become a bit of a joke.
I have some stock answers to that one:
1. If the eagles did fly over Mount Doom and 'drop in' the ring, then why would there have been a book at all.
This is your petulant reply, if you can't be bothered arguing any further.
2. The eagles have their own minds and they are their own race, and as such they are not at the constant call of anyone.
3. If the eagles flew into Mordor carrying The One Ring/Frodo then the Nazgul would rip them apart.
Mostly though, I don't mind questions or weird statements about LOTR, as they just give me a further excuse to talk or while the day away with e-mail chains about my favourite topic.