The movie sadly didn't made this very clear but I think Tolkien himself made a comment on that decades ago when asked about the eagles in his story. Don't know if that was mentioned in the books though. If I remember it correctly the point with the eagles was that they have been powerfull beings. They usually don't mingle in the affairs of >mortals<. Not on the level as Gandalf did who was send to Middle Earth next to the other Wizards with the specific task to help the people of middle earth fight Sauron. It all goes as far back as the Simarilion which explains the history of Middle Earth. The reason the eagles helped Gandalf apparantly is because they still owned him some favour or something. But he can't just summon them like they are his Servants. The eagles are ruled by Gwaihir. The King of the eagles. Now look how the Kings in Lord of the Rings often act. Even in the face of destruction and doom. They are proud and don't just do what you command them. Even if it's Gandalf. So Gandalf could probably ask the Eagles and their King to do it for them but it is very likely they will say no due to the risk of loosing to many. They are not very common in Middle Earth anymore. And as powerfull as they are they can be killed. So they keep mostly to them self.
However next to the points R.Graves made it only seems a good idea on the surface to just take the Eagles and fly to Mount Doom. Just get there quick, throw the Ring in. Right? Well. No. Because when you think about it Sauron would most likely spot them. He is searching for the Ring after all with all his power. And the Ring also wants to return to him. I imagine it would be fairly easy for Sauron to spot the eagles. Either trough Saruman or one of his Servants or what ever. And then it would be all of the power of Sauron, his Nazgul and what ever else he has against the Eagles. They wouldn't stand a chance. The whole point of the Journey was to get Frodo unoticed in to Mordor and Mount Doom.
