No. The plan would be revealed and they would have been intercepted. Keeping the plan (to destroy the Ring) secret was their best chance at victory.

The Eagles are not some Middle Earth public transport service and have a will of their own they rarely will interfere in the business of the other inhabitants of Middle Earth and don't follow orders from Man,Hobbit,Dwarf,Elf or Wizard(maybe apart from Rhadagast the Brown)The only reason why they helped Gandalf was because they owed him a favour helping them against some Orcs and The Eagles really hate Orcs.

The Eagles not being a public transport or not being willing to transport them is not a real relevant reason. I think that's not even mentioned in the movies, only in the books. And this is about the movies.

The important thing is exactly what Boromir told to everyone else. "You can't just walk to Mordor or fly to there".

Flying to Mordor would be suicide and ruin everything quickly. Remember that even way before they reach Mordor the skies were filled with spies let alone within Mordor.

Actually, it was the case. Gandalf wanted to go secretly to the land of the eagles, to the other side of the misty mountains compared to Rivendell. He kept the information secret for security reasons. To reach the eagles, they went through the mines of the Moria. unfortunately gandalf "died". But just before he fell, he revealed his plan by saying "fly you fool". Understand "Don't be stupid, do not go to the volcano on foot but by flying, with the eagles." Presumably it was not made explicit as goblins were nearby and could have heard the plan.