This family is going to eat healthy if it kills us.

No it is an exaggeration, that the parent has decided the family is going to eat healthy even if the endeavor is causes them a lot of work. The family probably eats a lot of junk or fast food and the parent has decided it's time to change direction nutritionally.
Generally, if you hear anyone ever say they are going to do something "even if it kills them", its a joke or a half hearted vow to change. You hear it said a lot around new years when people make resolutions to quit smoking, go on a diet or get fit. The phrase dramatizes their intention to do the right thing.