I feel uncomfortable when I breastfeed my newborn in front of my two sons (age 10 and 12). Could this cause them to have psychological issues in the future?
No. The older boys will be fine. Children adapt well to the natural order of things. Your attitude is more defining than the exposure....So if you are uncomfortable, be covered when you nurse (with a nursing blanket) or nurse in private. It is good bonding time with the baby and you can choose to do it privately. They will respect that. The more comfortable you are, the more comfortable they will be.
No. You need to do what you feel most comfortable with. You should have a quiet, relaxed place to nurse the baby. Ten and twelve year old boys can learn to respect privacy. But, the benefits of seeing and understanding how babies are fed will be invaluable for them as adult men and fathers. Explain your feelings to them, and what you feel is appropriate.