[icecube]

Originally Posted by Matata

The eagles used to be allied with the good guys (elves, dwarves, humans, etc.) and used to fight in their wars against evil (the armies of Morgoth). The eagles motivation for getting involved was that Morgoth had a pile of dragons which could destroy the eagle's nesting areas, and dwarves and elves were pretty good at killing dragons.

Sauron only has a handful of drakes, which aren't enough to pose a real threat to the eagle's nests and the eagles don't see the point in dying for someone else's cause. Due to their ancient relationship with Gandalf, they were willing to help him out, but only when he faced certain death. They did save Frodo at the end, but that was only when Mordor had been emptied of it's armies and Sauron utterly defeated.

Asking the eagles to fly the ring across a continent, at the same time Sauron was using the bulk of his might to find the ring, would have been a very tall order and very risky. It's unlikely that they could avoid the gaze of Sauron or his agents for the whole trip, at which point Sauron could intercept them and/or fortify mount doom, and end up with the ring.