This blog post comes from a forum post I made. The OP asks for help with his liberal (read: state socialist) friends. I responded in vast overkill because somebody got me going on something I had lots to say about. The core advice is not specific to liberals at all. It is advice for persuasion, period.
Don't be on the defense. Don't put them on the defense. You are not enemies, you are friends. Assuming you are both on a quest for truth, you are partners in the search for that truth. You help each other figure out what is true and what isn't.
Some people are not looking for truth. Some people are looking to preserve their identity as a liberal. If they value being a liberal and the relationships they would lose if they stopped being a liberal more than they value truth, then they're hopeless in that condition. You have to figure out a way to make truth their first priority. Do not let them lie about how truth is their first priority (the contradiction is obvious, and most strongly indicated in whether their decisions and actions contradict their words). Trap them with it if you have to. Don't be hostile or threatening to their worldview, but point out their hypocrisy as humor that they can laugh at (so they won't be angry at you). If you can align them with truth, and align yourself with truth, then you don't have an enemy, just a friend that doesn't know it yet. Persuasion isn't just about giving people truth, it's about getting them to accept the truth, which requires getting them to want the truth. Do not just appeal to facts, appeal to values.
From this sentence: "its okay to use the government gun when its for the good of the people", it is apparent that they value "the good of the people". If they are being honest in this (i.e. no ulterior motive), then showing them that universal healthcare is detrimental to "the good of the people" will be an effective tactic. But they probably aren't, they probably heard that rationalization for it and regurgitate it like a good liberal, where their true values lie: being a good liberal = being a good person, by the reasoning of many liberals.
It's actually a pretty common mistake for newer libertarians to make to discover how universally true the moral argument is, and then use the moral argument against people who don't share your moral premises. They value morality, in all likelihood. They do not value your moral laws, they value theirs. Arguing from it's immorality by the laws you propose will just make them disagree with your moral laws insofar as they do not already share them. The true basis for the morality of decisions is the results of the decisions. "Good" is what is good for people. This should be obvious because of all the pseudo cost-benefit analysis involved. If you asked them whether or not the costs of universal healthcare, without the benefits, were good or bad, they'd be an idiot to say that nonzero costs with zero benefits is good. The problem is that they believe that the good outweighs the bad. Fundamentally this idea is refuted by the fact that you will never get any more benefit from the government than it costs in the first place. At best it's hypothetically equal, but there's always overhead and bureaucracy and bullshit like spending social security dry, so outputs never exceed inputs, preventing it from being good, the costs always exceed the benefits. Bureaucrats and statists ALWAYS emphasize and overstate the benefits and ALWAYS downplay or underestimate the costs, using highly advanced deception tools developed by intellectuals recieving state subsidies or grants.
In cases where there's a negative externality, like aggression, the good of the aggressing individual is threatened by the retaliation of the victim who suffered loss. That's a good rule of thumb to use a lot of the time, if somebody wants to retaliate against you because of what you did to them, you probably costed them something. In the case of the state, there is no retaliation of the victim against the state, almost ever, so it can appear intuitively moral if it is believed that the act of aggression served the good of someone else. The moral costs appear to be virtually zero, and the benefits appear to be nonzero, so obviously universal healthcare should be good! The hard part is getting them to realize the unseen costs.
The issue with identity-value like wanting to be a good liberal is that if you bring up the costs, they subconsiously see a threat to their identity, and they'll say something according to their identity that makes them feel like a good liberal and that superficially appears to refute what you said, like "yeah, you should go to jail for not paying taxes". But that obviously just moves the problem back a little bit, because then you've got to repeat the process for the costs of paying taxes without the benefits, to get them to admit that there certainly ARE costs, because once they've acknowledged the costs they are forced to re-evaluate the costs and benefits. If they decide the costs of taxation outweigh the benefits, good going! If they don't, they're stupid, and they'll probably bring up more "i'm a good liberal" rhetoric or "you're a bad libertarian" rhetoric (they think they are good not only because they are what they are, but because they are not what they are not, and because their enemies are who their enemies are), and usually I'd get something absolutely irrelevant like "but who will build the roads?" or "best country in the world you should be grateful" or similar stupid shit, and if you can't get them back on topic after they try to do that, they're suffering from severe cognitive dissonance and you need to let them ferment and cool off because they'll probably be angry because you made them scared because you threatened their identity as a good liberal but they won't admit that to themselves.
This is why I prefer to either argue from the first principles and original facts, or to convince them that their first priority should be truth, depending on the the circumstances. If you don't have them wanting truth they won't care about facts and principles so you should really go for that first. I can't overstress it enough. People really do let their desired truth overrule the real truth. This is the number one enemy of reason and it's pretty much my definition of faith.