Most of us have experienced it ... someone is trying to cow you or impose their will on you through direct eye contact. They're relying on your instinct to avoid confrontation and look away, making you the "loser" of the confidence test.
Whether you think so or not, you've just given some interpersonal power to the other person. Whether it's a salesman, drill instructor, police officer, con artist, or cult recruiter, you're one psychological step behind in this battle of wills. (Note: it may under certain circumstances be wiser to let a police officer feel he/she has control over you. Cops are not as nice when they feel threatened or disrespected.)
This is a simple trick that allows you to disconnect from the uncomfortable eye contact without "conceding" the match.
Soften your gaze and direct it between and slightly above the starer's eyes. Try it on a friend and you'll see the difference — there's no connection, no feeling of intrusion because there's no true eye contact.
You can now converse more comfortably without giving your would-be dominator the satisfaction of making you break your gaze.
Body language: avoid excessive hand gestures, as they can be interpreted as nervousness. Hand gestures, when used, should be sharp, assertive and in a downwards motion. Don't show your palms to the other person, this is a submissive gesture. Watch politicians giving speeches to get the idea of the kind of hand gestures you should make ... they're professionally trained to appear confident and "alpha" at all times.
Final note: anyone (other than perhaps a police officer doing his duties) who uses nasty manipulative techniques like this is probably not a good person, and does not have your best interests at heart. The best reaction may be to stop talking to them, because they might have more tricks than this one.