I think it’s possible to make a broader and more straightforward conservative case. You start with one individual person, and their individual liberties, and you treat the imposition of requirements and restrictions on those liberties with skepticism. From that starting point it’s not that hard to understand how people can look at the result and say “they shouldn’t be able to make me do that!”
It’s probably universally accepted that you make public policy decisions using a kind of general balancing framework: is there something that is so important that we should make an exception to the rule “people can do what they want.” Like you look at the dangers presented by not having a minimum drinking age, and you ask whether that’s enough for the government to get involved. Most people agree it is, and then you quibble about what age and whether there should be exceptions, etc.
With mask and vaccine requirements, the imposition is definitely attenuated by the fact that it’s private business doing the requiring, and it’s attenuated even further by the fact that these are laws actually restricting the restrictions. So it’s not not just that you’re asking the government not to restrict your rights, you’re demanding that they restrict somebody else’s rights for you. But it still can be placed in the same general model, and there already are some examples of things the government says businesses can’t require. It’s not like the idea that the government would ever interfere in that kind of private affair is totally new and wild.
So it’s a judgment call about whether the imposition on my liberty, in the form of [allowing private businesses to require me to wear a mask] is justified by the dangers being prevented. And then another judgment call about whether that imposition is so bad that it needs to be remedied by the government prohibiting private businesses from doing it.
Obviously this all gets politicized, and you have to be somewhat flexible about which private rights are important and which ones are not in this context in order to arrive at the conclusion. But if you’re convinced that the danger of the pandemic is low, that the danger of private businesses being influenced by [some malign forces that are putting pressure on people to exaggerate the pandemic] is high, and that the danger of laws prohibiting mask/vaccine requirements is relatively low, you can imagine arriving at the conclusion that in these specific weird circumstances, a law that says you can’t force someone to wear a mask or be vaccinated to engage in commerce makes sense.
Just as a sort of thought exercise, if I personally believed that there was some kind of HIV hysteria, and businesses were going way overboard about HIV prevention for PR or some other irrational reason, I suppose I wouldn’t care if some jurisdiction made a law that said businesses can’t say customers need a (hypothetical) vaccine or to wear a mask to come in the store.
So, as long as you can come to terms with the fact that lots of people really do believe this is just a flu, or something less serious, you can pretty easily see how they arrive at this conservative case. Their priors are not the same as mine, but I, even as a relatively very authoritarian person when it comes to this sort of thing, can at least recognize the process.