No direct flight out of China. Are we still eligible for 144 hour visa?

No, unfortunately it is not allowed to have two mainland Chinese cities in a row regardless of the nature of those stops. When traveling on 144 hour visa free transit you are allowed to have a landing within just one transit area and then to proceed by a nonstop flight to a third country.

You are advised to try to alter your itinerary to be in accordance with 144 hour visa free transit rules or to obtain a transit visa (G) if you want or have to retain current itinerary. I'm sorry.

P.S. If you take a look, you will see that there are direct, nonstop flights from Shanghai to the Philippines offered by at least four different airlines.