remove the vinyl tile for sure and the question is, can you afford all this extra height if you leave the luan? This is your call, you have dishwasher considerations, of course floor transitions to other rooms, door issues, and then molding issues. Your call.
Secondly, I would thinset the cementious board down to the floor. This stuff is tough and screws in the kitchen might rust and give way. Be safe, better than having to go back in later on.