I think EU law steps in here. Here in Ireland, you are entitled to take a force majeure day if you are caught in an emergency with someone to whom you can be classed as acting in loco parentis to, and this stems from an EU directive on labour law.
This does not cover babysitting though - and is only applicable on dayone -if you get someone, say your child, to a hospital, then the force majeure is no longer applicable. Similarly for a sick child, you can't expect it to cover the flu.