McConnell has opposed it every time it comes up. I used to think this was a good idea until I heard McConnell explain his opposition to an amendment. He said "where does it stop? Should we also pass an amendment banning Roseanne Barr from singing the National Anthem?" If you recall, she sang it once in a VERY disrespectful - disgraceful - way.
Scalia, the most conservative of the SCOTUS justices is also against the anti-flag burning law...though I guess I've never read or heard him speak regarding the Constitutional amendment aspect of the same.