If the child of a dual-military couple gets put on quarters, does the soldier get quarters too? Can the unit force the soldier to take leave?

This is why a Family Care Plan is required. Emergency caregivers are required to be identified for just such an occasion

Now, as a leader, if I can spare PVT Schmedlap today, and his kid has been kept hope sick from school, then I will let PVT Schmedlap go take care of kid and check in periodically throughout the day via telephone. But only if I can spare him. And only if it doesn't become a habit.

Rules may have changed since I retired but.... The child isn't in the service so kinda confused how they are on 'Quarters' restriction. Maybe the medical provider intended for the child to not attend school due to potential spread of a virus/contagion/disease/etc. A one/two day pass might be approved given circumstances but otherwise I would surmise the service member would be expected to be at their appointed place of duty unless they use available leave time.