Why didn't the eagles just fly the ring to mount doom instead of making frodo and sam walk?

No, secrecy was not required because they were walking. They were walking because secrecy was required. Also. Remember, Gwaihir was only going to get them as a favor to gandalf. Sending them on a mission of death (which is what it would have been if they'd attempted to enter while the will of sauron survived inside the ring) would have been too much for gwaihir to ask of his brothers. They would not have gone even if gandalf had asked. He did not have the ability to summon them (as someone else pointed out that radaghast sent gwaihir to bring tidings to gandalf). If sauron had suspected their intent was to destroy the ring, he would have killed the eagles and anyone riding them. Don't forget he had more powerful devices than his orcs or even the nazgul. He and gandalf were both of a race of demi gods. You don't think a god could stop a couple of eagles from entering his land? He blackened the skies for cripes sake. No. Sauron would have killed the eagles..even if his servants had been unable to (which I totally think the nazgul would have taken them while sauron still lived). Then he would have recovered the ring and good bye middle earth.