Do I qualify for the 144-hour visa-free transit in Shanghai?

Yes, but your journey has to begin in the country like US and after that your first port call has to be a country or territory different from both US and China, i.e., US - Shanghai (China) - Japan is fine and also US - Shanghai (China) - Hong Kong. You can, if you want have even more overnights in Shanghai if you wish. This is 144 hours visa free transit...