Well, government intervention should come in the form of laws. In the first situation a crime is being committed so it is clear cut.
The second one is a bit messy because the kid probably isn't able to defend his or her own liberty independently. The kid isn't really standing on his own principles if he refuses treatment because of the influence his parents have on what he believes. As a country we have decided that kids are able to think for themselves at 18 years old. Until then, mom and dad are in charge. Should the government intervene in order to increase the overall health of our country? Maybe. Who decides what situations call for government intervention? Well, we could pass a law and fox news could cover the court case of the family that is suing for the right to not be forced to act against their beliefs...more wasted money. Can you imagine what would happen if we forced Christians to sin for the sake of public health until they were 18? Then, at 18 the government said, "Okay, feel free to stop sinning whenever you want now that you're 18." They would collectively shart and I wouldn't be all that suprised if most of the government was assassinated.
So, to me, the only option is to arm people with as much information as possible and then let them make their own decisions. Where this becomes sticky is when various groups begin tainting the flow of information with bias. For example, Catholics want sex to be associated with sin and pain and burning in hell. So, why would we want to promote a drug that tries to prevent God from punishing fornicators? Or, maybe Pfizer decides to publish only the one study that shows that the drug is effective while the other 10 studies that show no statistical effectiveness go unpublished.
If we're going to use an approach to public heath that is so penetratingly utilitarian that we take away personal choice I suggest sterilization. To me, the way to get dumb people, or whomever the target is, to stop having children, or whatever the undesired action is, is to convince them that it is in their best interests to not have more kids. Then, they make the choice to have more kids or not to have more kids based on information they're armed with. Similarly, in 2007, abortion should have long since gone the way of the Apple IIe. It should be a non-issue because birth control and safe sex should be well promoted.
The end result will be that Americans that are paying attention should be pissed that they are REQUIRED to get their daughters vaccinated but realize that there appears to be good data supporting the vaccination.
side note:
The problem with calling other people's reasons illogical is that logic regarding beliefs is almost entirely subjective. Religous extremists think that everyone else's beliefs are illogical. Who is right? Me. Prove that my religious beliefs are wrong and yours are right. Or, since we're all scientists, prove the theory that my religion is wrong, right (H null).
Didn't jesus walk on water? Well, that is physically impossible.