If you wanted to get technical with it, you could make your buttons optical. You might be able to use optics from old mouse boards, since they are nice self-contained emitter and detector that would probably give you a logic signal with a few resistors.
But at that point, it might be less effort to get quiet buttons that don't click. I've seen some at Radio Shack (shudder... expensive... evil... but necessary...) that don't make any sound when pressed, but they are not the same size as the pushbuttons you already have in your panel. Could be mounted to button plugs, maybe, for a quick try.
Can you just edit out the very beginning and end of the recording, where you push the record and stop buttons? or do you have to be pushing buttons all throughout a track?