Yes, it is possible for you to cause allergy eyes in the sun. The strong sun lights contain a lot of uv rays which will be bad for your eyes. That is the main reason of your allergy eyes. You should wear the sunglasses in the sun to prevent the allergy eyes. If you get the sunburned eyes, you could use the warm compress to make your eyes feel comfortable.
Well, I am afraid that your assumption could be possible, because according to an awful lot of cases that some people would be allergic to sunshine or strong lights, and they would sneeze or shed tears. Anyway, this is primarily resulted from decrease in their immune system or some eye diseases. So, it would be wise for you to have a medical checkup to find a better way to deal with the problem. Also, try to build up your immune system by taking more exercise and have a healthy diet.