Maybe you are allergic to something or maybe you got an eye infection from the puppy. Have you touch your eyes without washing your hands after playing with the puppy? If you did, I think the problem lies in the puppy. You can have your eyes checked by an eye doctor. If he or she told you that you are allergic to furs, you can prevent this by avoid animals and other furs. If he or she told you that you got an infection. You should take medicines to help with your symptom.
You can't rub your eyes with your hand. Go and wash your hands with soaps carefully and then apply some artificial tears into your eyes. You can keep blinking your eyes because this can produce more tears to moisten your eyes and wash out foreign objects or odour that caused your eye irritation.