The thing with Reggie sounds like inverted, or inverse sneezing. Many greyhounds seem to have this problem. It is thought to have something to do with allergies, so it totally makes sense that he had this problem when you were camping this weekend, and he was sleeping outside in a tent.
When my Buddy has an inverted sneezing attack, we tilt his head so that his nose is pointed up, and gently rub his throat, and talk softly to him to calm him down.
It's kind of scary sounding at first, but it's not a life-threatening problem! (You might want to talk to your vet about allergy medication for Reggie, though, if the problem keeps persisting!)