Why do you want to stick with square tapers? That frame probably has a BSA threaded BB that is 68mm wide. You could put most any two piece Shimano road crank on it. The BB is inexpensive and easy to change out with just a few special tools. You do realize you'll need a front derailleur and STI for it. The FD is inexpensive, but the STI will cost some depending on what you get.

Contacting HUPcc might get you the answer of whether or not the more normal size 2 piece Shimano cranks will rub on the stays or not..

Your son may not fit that bike too much longer. So consider how much you want to put into that size frame.