No, Allergies do not strengthen your immune system. An allergic reaction occurs more or less because your immune system confuses a non-harmful, foreign substance that has entered your body system with something that is harmful. There are signs that strong immune systems can help aid in fighting back against and possibly preventing cancer, but in this case, simply having allergies isn't going to make a difference in your immune response. Additionally, your likelihood of getting cancer is not as much based off the strength of your immune system than it is on other factors, such as lifestyle choices and genetics.