Do I qualify for the 144 hour Visa Waiver?

Hi!! I am planning a trip to Beijing in July. I'm a U.S. citizen. Here is my plan:
DFW-PEK (for 4 days)-HKG. Would this qualify?

Also, after spending a few days in HKG, would I be able to go back to BJ if I wanted to and use it as a transit again and use the 144 hour waiver, or would that not qualify?

Thanks so much for your help.