I don't think anybody is proposing requiring all vaccines. There are some vaccines which kids clearly shouldn't be required to have, like say, typhoid (though if parents were smart they'd give it to them if traveling to certain places). The thing is, which vaccines should be required has already been determined by most states. The problem is that parents can simply fill out an exception form in lieu of getting their kids vaccinated, and that's what needs to be done away with. The only reason on the exception form should be if they can't get the vaccine for medical reasons (e.g. allergic to it), which must be accompanied by a physician's signature (a real physician, not a naturopath or chiropractor).
As far as chickenpox vaccine, I do think that ought to be required. My kid shouldn't have to get sick and miss a bunch of school just because you don't want to pay $40 for the vaccine, because even if I vaccinate my kid, that particular vaccine is only 85% effective. If everyone is vaccinated, the chances of my kid catching it are very small (it's not just marginally effective if everyone gets it!), but if others don't get my kid pretty much has a 15% of getting it. Plus, I bet most parents spend more than $40 when their kid gets it! Doctor visit copay, medicine for the fever and itching, chicken noodle soup...etc (not to mention missed work for many parents!). So yeah, that vaccine should be required.