That's not entirely true.
I've been vaccinated against chicken pox my entire life and have never had either the chicken pox or the measles. Remember, once you get Chicken Pox the virus never leaves... it just becomes dormant so that later, as an adult, it can reactivate at a time when your immune system is compromised and give you a nastier disease called Shingles. Shingles is what occurs when the virus starts trying to eat your nerve endings. It is better to try and avoid getting the pox to begin with and get the vaccine.
Having said that, I do understand your fear. When I was four I cried so hard over the prospect of getting a needle that I woke all the babies in the centre and I was taken home.
This is something that you will either grow out of or be scared of your whole life. Best advice I can give you is not to look at it and try and distract yourself. Remember, the vaccines are there for a good reason and the temporary pain is better than the long term diseases.