Aristoteles said that man is a social animal, we should never forget our roots, they explain a lot of what we are. Flame and Thef0x wrote good posts. The problem is not about caring what the others think of you, it is about caring too much. If you manage to raise your self-esteem and your overall confidence level, the problem will solve by itself. As for the practical ways to do it... It really depends on your situation.
As for moviestar :
In human psychology, a deep prolonged eye contact makes people unconfortable because it is interpreted as a sign of defiance (especially between men, but you will make a woman to feel insecure as well if you hold an eye contact more than 7 secs if I recall correctly). Also, women eye contacts are clearly an invitation to approach sent to the males, so I'd avoid it if I were a girl (they do it naturally anyway). It is hard wired in our brains, so I'd be careful with your exercise, you never know who you're looking at
