Well, of course not. It will not be bad to have reading while eating. First, you should know that reading and playing video-games can not be bad. On the contrary, the inactivity is bad. Generally speaking, what you eat will have an effect on your health. Obviously, if you eat a good low-fat, high protein meal then you don't have to worry about anything, but if you've just eaten something fattening and then sit down and do nothing afterwards, it can just be harm to your health. But you should remember that there is nothing to do with your eyes. But on the other hand, your sitting position will influence your sights, in some degrees. So just sit in a right position. Also, the light will also be important.
Yes, reading books while eating is bad for your eyes. You could not focus on the words in the books because you have to use some energy to eat. Thus, the eyes will easily get tired. At the same time, the words may not fully come into your minds. Thus, you'd better listen to your mother's advice.