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So, Connecticut wanted to join several other states in ending the religious exemption to childhood vaccination requirements, and thousands of people showed up to protest the bill at a hearing that lasted 24 hours.

Okay, I suppose you want to know how I feel about this.

First let me get a couple of points out of the way that are specific to this issue but not really to the principles involved. The only, for want of a better word major religion I can think of that would actually forbid vaccination is Christian Science. That doesn't really matter, however, because people can make up whatever religious beliefs they want and even get together with their friends and give it a name. "Church of True Believers in the Immaculate Body of Christ" or whatever.  We can't really go around deciding what does and does not constitute a religion and furthermore, being in a minority doesn't mean you have fewer rights. Also, the absurdity of your religious beliefs doesn't matter, since all religions are equally ridiculous, though some are more equally ridiculous than others.

The general problem here is the inevitable problem of balancing liberty with general public or social welfare, whatever you want to call it. This case also includes the subsidiary case of parental rights over children. An absolutist libertarian position is trivially absurd. If your religion requires human sacrifice, you cannot practice it in the United States. The same goes for any violation of criminal law. You can't claim that your religion exempts you from the Clean Air Act, traffic laws, or for that matter taxes. By law, religious organizations don't have to pay income taxes, but that doesn't mean the law couldn't change. The question of what constitutes a real religion does come up here, because again, anybody can claim that there house is actually a house of worship of some kind so the courts do apply a test of sincerity but it's damn tricky. So the freedom to practice religion does not extend to violation of ordinary laws.

Perhaps oddly, however, the Supreme Court has ruled that Native Americans have a right to use peyote in religious practices, which is otherwise illegal. Not being a lawyer and not having studied the issue, I can't tell you offhand what their rationale is, but at least the practice is not evidently harmful to anyone else. Which means peyote should be legal for everyone, right? Anyway . . .

Parental rights over children are also constrained. You can't torture them or starve them or rape them, or for that matter deprive them of education although home schooling and religious schooling are not very well regulated in many states.

So refusing to have your children vaccinated clearly endangers other people, including the children themselves and people they come in contact with who, for medical reasons, cannot safely be vaccinated. Your religious objection is objectively preposterous and has no conceivable basis in logic or spiritual experience. It's just ignorant. Therefore you do not have a right to do it. End of story.