A.P.
I wouldn't worry about it. As far as eating "problems" go that is a good one to have. Broccoli and tomatoes are nutritional powerhouses as well as fruit and brown rice. Breads-not so much. So while it may not be very convenient for you for eating out, rest easy in that he is eating better than most kids. My sons really eat no veggies but will "feast" on calorie dense, nutritionally deficient white buttered bread like there's no tomorrow!
I will tell you that you should keep offering it to him. When my kids refused foods when they were little I let that be the end of the line and really didn't push it again. Experts say that you should keep offering as it may take 10-15 trys before a food tastes good. Wish I would have done that when they were little like yours.