He's rejecting being labeled (and dismissed) as a "born-again" type - he's not rejecting Christ, whose words he quotes approvingly in the same interview - in fact in the very same answer. Dylan has always considered labels to be radioactive, because they mean such different things to different people. You or I might hear "born-again" and think, hey, that's great. Someone else hears "whacko nut-job." Dylan is very sensitive - you might well think perversely so - to not letting labels overwhelm truth.
Born again (by the Spirit of God) believers know the Truth and can spot the false. That interview convinces me beyond a doubt that Dylan is not a born again believer in the Lord Jesus Christ, at least not at the time of the interview.