Flying I’m from Dubai to Beijing for 3 nites can I take train to Shanghai & fly out from there?

According to the policy, you are not eligible for 144-hour visa free if you want to travel from Beijing to Shanghai. 144-hour visa free is applied to the passengers who travel from a country to another country and transit in only one of the permitted areas in China, but Beijing and Shanghai belongs to two different areas separately. However, if you want to travel from Beijing to Shanghai, you need a Chinese visa.

So if I fly from Dubai to Beijing and then return to a third country only THEN I can get the 144 hour visa?

Yes.