From US to Shanghai with Delta, Shanghai to Hong Kong with different airline. Will this work?

Yes, Hong Kong is an eligible territory as far as 144 hour visa free transit is concerned and separate tickets as well as unrelated airlines are fine and accepted.

So, if your itinerary looks exactly like this:

US - Shanghai - Hong Kong

And you do not have any landings of any kind between US and Shanghai and Shanghai and Hong Kong, you are definitely and undeniably eligible for 144 hour visa free transit in wider Shanghai area.

When you land in Shanghai the first thing you will do is to approach 24/144 transit counter where you will apply for a temporary entry permit by presenting your flight tickets and your accommodation booking confirmation. After a short check, your permit will be granted and you will be allowed to cross the immigration and to spend your time in Shanghai and its wider area on your own. Have a nice journey.