Filipino citizen married to UK citizen, both US Green Card holders, traveling to France - visa needed?

Based on what you have told us, you and the children will need

1. A Schengen Short-Stay Visa (or an EU Family Permit see comment from 'phoog', to whom thanks, below); and
2. A UK Entry Clearance (Standard Visitor Visa)

This is because Philippine nationals require a visa in both regimes. Having a US Green Card is not recognised by either the UK or Schengen as a visa you can travel on, but a Green Card is always helpful in the application process. Being married to a Brit can sometimes be helpful in acquiring an entry clearance, but does not remove the need for a visa. Without the correct visas you will not be able to board the aircraft.

Repeat: you cannot travel to the UK or the continent on a Green Card.

They will stop you when you try to check in and refuse boarding. This has screwed up more people than you can imagine, including businessmen and businesswomen who should have known better.

You would complete the Schengen application first and if it's successful, then go for the UK application.

There is no 'family application', and each person must have a separate application and pay the fee. There is a 'best practices' strategy for sequencing the applications, you can read about it here. And here.

See also: How should a family with young children apply for UK visas?

Now about the children, if your husband was born in the UK and he is the biological father (or has valid adoption certificates), then you should register the children with the British Consulate General in New York and get them British passports. Contact them and they will know what to do. Doing this will reduce the cost of visa applications and make life more convenient. Of course if your husband is British by descent, then this is not an option.

Note in passing, since you speak about "take a Eurostar train from London or Dover to France", that the Eurostar trains do not go through Dover. The Channel Tunnel begins at Folkestone, some 10 km west of Dover, but Eurostar does not stop there either. The last passenger halt before France is Ashford international, but only 3-4 Eurostar trains a day actually stop there. In the vast majority of cases, your most convenient boarding point will be London St. Pancras International.