Is it appropriate for the "greatest fans", no not at all. It directly opposes, IMO, being a fan in the first place.
Sure Nebraska fans are great and all, but lately (the last 2-3 years or so), I have come to realize that when it comes down to it, we are pretty much like any other group of fans. We have some people who think they know it all and could solve all the teams problems looking from the outside in. These guys boo. Always have, always will. We also have people who are just fans of the program, plain and simple. Sure, they get disgruntled when things don't go well but what fan doesn't. That doesn't mean you add to the bad play/game/whatever by booing your own friggin team!
As far as booing, I would never really boo my own team at a game. It doesn't make sense to me and can do nothing to help the team that you want to do well.
I agree totally with both of you. What purpose does booing your own team serve.