Is there a proven way to trigger an allergic or allergic-like reaction in non-allergic people? I have wondered if this is possible, since allergies are basically just a form hypersensitivity - so could this be triggered somehow?
At least, not that I've ever heard of. Allergic reactions, as you point out, are an overactive immune system response to an allergen (anything that causes the reaction). What determines who is allergic to what is a mix of genetic and environmental factors, and because an allergic reaction is categorized as an error on the immune system's behalf, inducing an allergic reaction is essentially finding a way to force the immune system to make a very, very specific error.
What can (and does) happen is the development of new allergies. Almost all allergies are acquired after birth, and repeated exposure to substances can lead to the appearance of new allergies - one case that stuck out in my head was a woman developing an allergy to a copper IUD that she tolerated well when it was first put in place.
I am not aware of any ability to predict what substances will cause the development of a new allergy. Both the unpredictability and inability to induce allergies comes down to how the immune system reacts to antigens (foreign substances). The machinations of the immune system's response to an allergen or antigen is fairly complicated, and at any point in the chain of events the potential for an overreaction is there. That's probably another question, though, and a much longer answer.