in the movie, they have those flying nazgul(don't know the name) things so it's obvious that they couldn't do this.
I haven't seen them movie in so long but there was that scene where the flying nazgul thingies was flying around thats why they kept hiding on the rocks and bushes.
and the author of the book also explained that the eagles isn't in a direct command from Gandalf, They simply helped.
In the books we have seen them appear and helped maybe five times, like in the hobbit where they helped the dwarves escape from the orcs, they helped because the eagles hates orcs not because they were ordered too.
