So what can you do? Actually there is a lot you can do, and answers fall into several general categories.

Retraining The Reflexive Body

One of the best drills is to use a pad (often on a handle) to simulate various attacks -- straight punches and hooks to the head, body shots and various types of kicks. Try to keep a proper fighting position, hands up, and move -- shift position, duck under hooks, and shift your head to the side of a punch (known as slipping a punch) while blocking it with your hand or forearm. Gradually you will become more comfortable with being attacked and the pace can be increased. The pace of the practice attack should always push you a little beyond where you feel comfortable. This forces you to improve.

The Mental Switch To Offense
Most beginners focus on defense and not getting hurt. Mentally they are defensive and physically they are hesitant. They are waiting to get hit or hurt, and when anything comes near them they jump in anticipation.
Try to reorientate your thinking to one of offense, not defense. Mental focus will change from avoidance of getting hit to a focus on attack. You will begin to look for an opening and your dialogue will change too: "Good, here comes an attack I can respond to." Fear is reduced because attention has shifted elsewhere.

Visual Reorientation
Another method to reduce flinching, blinking and turning away is to be able to re-educate how you use and control your eyes.
To reduce blinking you can control the muscles of the eyelid.
Many mistakenly try to hold their eyelids open to stop blinking, but this doesn't work because blinking is done with the muscles that close the lids, not open them.
Thus to stop or reduce the tendency to blink, slightly narrow your eyes, thus bringing conscious control over the muscles used in blinking. Have a partner punch to either side of your head, or jab toward you face but at a distance that contact won't be made. Practice controlling your blink reaction. Keep your eyes narrowed, observing everything, but not focusing on anything.