Do you have to look at your oponent eyes when you handshake them?

I'm not sure about what is considerer appropriate etiquette. 

I was raised to look people in the eye.

If you want to be show how socially inept you are, do not look them in the eye. Seriously though, why wouldn't you? Holding out your hand while looking somewhere else is a little odd don't you think?

I think some opponents would take advantage of this to scowl at you in order to make you lose focus.

If a scowl is going to make you lose a game of chess, then chess isnt for you.

The fact that I'm even near another carbon-based life form is enough social interaction for me. So no, I do not look people in the eye nor do I talk or touch them.

The thing is that handshakes are mandatory according to the rules.

It is not needed, you can choose to look at them or not, smile or not.

If you can't cope with a scowl at a chess game, you deserve to lose.

You sound like a real fun guy to be around

That's a strange specialization. 

Well you failed. You've never even seen my face let alone my eyes!

I think the Pros do this; and it is useful for you to copy them:

If you are white, you can look down at the chessboard to move your piece and hit the clock while shaking hands.

If you are black, you can look down at the clock and hit it to start his time while shaking his hand.

sometimes you get wrecked so bad that you can't look the other person in the eye:

I can't even see their eyes. Quality photos only please.

Morozevich (on the right) is looking at the board while Anand (on the left) is looking directly into his eyes.

the picture is from 2007

lol nice