There is a type of meditation called Trāṭaka. This style involves looking at a fixed point such as the flame of a candle.
Eye meditation is a subset of Trāṭaka. In order to do this, take a mirror and begin staring at your eyes. In my experience the best way to do this is to take a few breaths to relax and get used to your image in the mirror. Look into your eyes, specifically focusing on the inner iris. Do not try to hard. Relax into your gaze and just stare. If you break your gaze, relax and try again. Focus very diligently on your inner iris.
This style of meditation can allow you to become more confident at looking people in the eye while talking with them. It is also relaxing and can help you feel more connected with yourself.
Let me know what you think if you decide to try this out.