Hello there! To tell you the truth, I'm not sure that you square the variable in order to normalize it (if that is what you want). In such case, you could simply take a log and use it in the model (the interpretation of the coefficient would be different, though). When you square the variable, you assume a curvilinear relationship between X and Y -- and that is a quite complex issue that is heavily dependent on theory. Besides, if you indeed assume curvilinear relationship and want to test it, then both X and X_sq must be retained in the model.

Hope this helps.