144 hours free Shanghai visa

With US passports, we will travel round trip from Seoul to Taipei.
1. From Seoul, we will stop by Shanghai for (lay over for 2 nights, leave airport, rent hotel), then to Taipei.
2. To Seoul, we will stop by Nanjing (lay over for 13 hours, stay in airport, no hotel), then to Seoul.

Do we qualify for 144 free transit at Shanghai?

Thank you.