Not a lot of information to go on, but the airbag is designed to deploy if the sensor detects what it considers to be an impact. If you did indeed have a blowout, as opposed to a puncture, then that may, depending on the precise circumstances, have created enough of a signal to the sensor to cause it to deploy the bag. If so, it has done what it is designed to do, so I don't see where any claim against a manufacturer would come from.
I'm not aware of any legislation that prevents you driving a car once the bag has deployed, provided that the loose material does not affect your view or your ability to use the controls, but I personally wouldn't want to be driving one with the dust that comes out of them still in the car. Take it to a garage or a dealer to find out about getting them replaced, preferably a main dealer or a knowledgable independent for the make of car as they will have the greatest knowledge about whether there have been issues with the brand.
