Travel from Houston-Shanghai-Hong Kong-Houston

Hi there! I'm traveling from Houston to Shanghai and staying for 3 days in Shanghai. Then I will be flying to Hong Kong and staying for 4 days. Does Hong Kong qualify as a 3rd country or region under the 144-hour free rule? I called the Chinese Consulate in Houston and they think it is best for me to get a visa, but I dont see any point in paying for an entry permit if I qualify for the free one. Any help is appreciated.