I have a fairly new Music Man John Petrucci BFR 7 string, and it has this weird problem where I can hear a high pitched ringing sound whenever I play the low E string. It's the most noticable when palm muting.

I already think I know what the problem is, the nut is not muting the string around the headstock, it's still ringing out. I've heard of people using foam and stuff to mute it, but is there a permanent fix I can do to the nut? It's a beautiful guitar and I don't want to put foam or anything on the headstock and ruin it. Thanks!
That's what I'm doing right now. The only problem is that it takes like three hairbands for me to fully mute it, and it looks pretty bad!
maybe a nut replacement is in order? should be relatively cheap compared to the cost of the guitar.
I hear this on all my guitars. I just use a hairband, as suggested. If you get it to cover all the strings in the right place, none of them will ring, with just one hairband. Just experiment moving it, between machine heads and different angles to where it mutes them all. I do this regardless of nut on the guitar as the nut seems to make absolutely no difference to this problem.