Is traveling to Shanghai followed by Hong Kong the to Phuket using the 144 free china visa

You are perfectly fine to do the as you plan, so your itinerary:

X country - Shanghai - Hong Kong - Phuket

Is definitely eligible for 144 hour visa free transit in Shanghai, but you should be aware that you have to arrive to Shanghai from a country/territory that is different from Hong Kong and you are not allowed to have any other stop within mainland China of any kind. In other words your flights from country x to Shanghai and from Shanghai to Hong Kong have to be a nonstop flights. While in Shanghai you are allowed to explore Shanghai city as well as the provinces of Jiangsu and Zhejiang.

In the end, you are advised to print out your flight tickets and your accommodation booking confirmation for, at least, the first overnight in Shanghai and to have those documents at hand. Also, please, use correct terminology because this is 144 hour visa free transit and not 144 free China visa... There is no visa at all.