The only person I can ever make eye contact with is my girlfriend, and sometimes I can't even manage that. Someone -- I think it was a previous partner, I can't really remember -- tried to make me make eye contact with them, but my eyes just kept sliding away, it's just not natural to me. Quite often I've been told to make eye contact with people, normally when I was getting told of by pedantic crazy teachers a few years ago, but it just seemed way too personal to do.
It's either a challenge, or intimate to me, and I'd not want to do either of those things to a random person, or even a good friend.