Can Dragonslayers eat their own magic?
I think they can, but they just wouldn't gain any energy like they would from other sources, so it's just completely pointless. There have been points with dragonslayers in pretty tight pinches, and I'm pretty sure one of them would have at some point eaten their own magic if there were any point in doing so/ if it's even possible. 

I think Natsu was just making a bit of a stupid analogy when he said it was like Lucy eating celestial spirits, at no point has there been any suggestion that a dragonslayer's element is in some way alive, and he's more than happy to fire off bursts of fire that presumably must burn out eventually, I don't think ethics come into it at all, just practicality.