I always initiate approaches without getting an eyecontact first. Eyecontact is an extrovert characteristic in my opinion, and an introvert girl will not show it unless she is really comfortable. That doesn't mean the girl isn't interested or not capable of a potential lay/relationship, it just means you have to do more work.
If she's shy, and nervous, it prolly means that she's not used to being approached by guys and doesn't know how to react properly. What you have to do, is make her feel comfortable. Have a good laid-back conversation, throw in some light humor, and get her to open up. Just make her feel comfortable, and then do a number close.