An aspect of this phenomenon (though not the entire explanation): Conservatives tend to oppose initiatives that are broadly beneficial - especially if they stand to help marginalized groups - if they require even a minor or suspected sacrifice on their part. In this case, inclusive of having to sublimate oneself to a public health campaign/the purported risk of side effects or autism, respectively.

I've also noticed that conservatives tend to have a pronounced reaction to the prospect of ills that have visited marginalized groups also becoming their problem. The US has a limited but sharp history of medical exploitation, particularly of minorities. Many conservatives think, "We're next, unless we resist forcefully."