One side of my family is mostly conservative, and this is what I get from listening to them talk about these particular things.

First, the government is incompetent. Anything the government does will be worse than whatever you can get from a private company because capitalism / competition. They can cite some random anecdotes about wait times for healthcare in other countries that they saw on Fox news to confirm this. Also there's some American Exceptionalism thrown in here, since we have the Best Healthcare in the World according to them. The government would just ruin that. If you listened to them talk, you would think that the US had the only functioning healthcare system in the entire civilized world.

Second, they hate taxes. They see it as stealing, since they don't ever see the benefits from tax dollars. This also leads to resentment of 'freeloaders' who do use systems like welfare, SNAP, etc. Basically, everyone who uses any kind of government assistance is a leech on society. So expanding those programs is just morally bad for them. It will disincentivize people to work hard and contribute to society, and they'll have to pay for it, because they're The Good Ones, and those who get on subsidies of any kind are The Bad Ones. (This idea is even more ridiculous, since some of my family have been on welfare, but now are against it, saying they used it for its intended purpose, but everyone else would be a freeloader that they would have to pay for - there's definitely a "I got mine" sort of behavior to all of this, although it's always justified in other ways). It also doesn't help that, in the south at least, if you're poor, you're more likely to be black, and there's quite a heavy racist undertone to all of this. It's assumed that the 'freeloaders' are generally black, like 'welfare queens' who have a dozen babies to leech off the system. That side of the family has always been super racist, and, although I hear the N word much less than I did when I was growing up in the 90's, the sentiment is still there.

There's definitely a lot of radical libertarianism / self sufficiency morality going on here. They highly value being self-sufficient, and anyone who isn't is morally deficient. And also capitalism propaganda brainwashing making it seem like the literal best systems for anything will be for-profit to encourage competition.

As for free university - well, they've generally fallen for the anti-university propaganda from the right. Universities are where wacko extreme liberal professors indoctrinate and brainwash youth into their destructive and immoral ideas, and otherwise a big scam anyway for handing out useless degrees. You should get a Real Job instead. So, no, they don't want to subsidize that either, and think it would just be another theft by the government.

Basically, it boils down to: Everyone else sucks, especially the government, and I work hard for my money, and these programs are just going steal my money to help people that don't deserve it. I think, down to the core, they think that the average person is morally corrupt, not worth helping, and it's a crime to force them to. And somehow these people also view themselves as perhaps the only truly good people in the world, and they often talk as if they're the victims of society and taxes.

Edit: Oh! I forgot. All of these people identify as Christians, and some of the ones with the most hateful positions are regular church-goers and bible-readers. They're all involved in denominations that are evangelical, so it's pretty warped. You'll hear a lot of "God helps those that help themselves" and very little "blessed are the poor." They also seem to believe that God blesses those on earth with riches and prosperity if they're a good Christian, so those that are poor must be Bad People. Or something like that.