When I were a lad, I were taught to look folk in the eye when they were talking to me or when I were talking to them. I was taught it was bad manners to do otherwise. So shutting my eyes seems ill mannered to me, and personally I avoid it. I've seen performers with their eyes shut do a good enough job, but as a member of the audience I would consider they appeared better performers if they had their eyes open, other things being equal.

If people need to shut their eyes to avoid being 'distracted' by the audience, then I find that strange. I admit I have been really distracted on occasion, but only by unusual events happening in the audience, but those occasions are very rare. A sea of faces just looking at me and listening to me I don't find distracting. Maybe people who have to shut their eyes shouldn't be performing in public if they feel that way?

My only problem is deciding who or what to look at - I tend to look around as I'm singing, but a lot of the time I kind of stare into the middle distance at no-one in particular, as some people probably don't like being stared at (unless they're people I happen to know). Maybe shutting your eyes avoids that problem too, but I don't think it's right.