There are two issues here:
Will the Canadians let you board an international flight without a passport or NEXUS card. The answer here is pretty definitely no. The airline company could get fined severely for doing this, so a gate agent is not likely to let your son through -- unlike for domestic flights where they might take pity on a kid.
Will USA immigration let you in without a passport or other government-issued documentation such as a REAL-ID driver's license or NEXUS card: Usually, but expect a thorough and perhaps delayed vetting process. They are required to let American citizens in, but they are allowed to hold you in a holding cell (for hours or even days) until they're satisfied that you are a USA citizen or refuse you if they are not satisfied. Anyone can photoshop an image of a passport passport or birth certificate on their laptop and verification can take longer than you might expect if the DHS and State computer systems are feeling ungracious towards each other, which can happen.
Given the latter situation, your son's best bet if he can't get an emergency replacement passport isat the American consulate in Canada, is to go to a land crossing and try to cross the border to the USA there. Tell him to bring as much documentation as he can. ExpectTell him to expect a very long delay so he should eat a hearty meal and go to the bathroom before presenting himself (and carry snacks, water, and a physical book to read). Sometimes they'll let people through in 15 minutes, but these days it might take much longer.