Can I go from USA > Beijing > Hong Kong > shanghai > USA?

Yes, you can if you have the connected international flights tickets and won't make any stop in other areas or cities of mainland China. Moreover, in your first route, you are only allowed to move in Beijing, Tianjin, Hebei and leave for Hong Kong from these three areas. On your second route, you are also only allowed to move in Shanghai, Zhejiang and Jiangsu and leave for USA from these three areas.