[paul544]

It has often been speculated why the eagles never took them. It was something Tolkien, as far as I know, was never asked. The obvious answer is the same as 'why didn't they just take horses?', Sauron would be on the look out for such things.

[Vast Buffalo Wing Conspiracy]

I think Sauron would notice an eagle trying to "sneak" into Mordor. It's kind of hard to hide when you are flying in the air.