Traditional insurance will/can reject some individuals.
For example, you don't take an insurance against theft AFTER you've been stolen hoping that they will reimburse you.

In addition, insurance companies can target a certain demographic to offer lower prices:
An insurance for office workers will not be the same as an insurance for football players.

This is a really key point. Market forces push insurance to offer specialised coverages that match your risks. Start smoking? Hey, we're going to bump you in the next bracket...

And in case the only insurance offered are a clear rip off (extended warranty kind of thing), you can always walk off and say "no thanks".

Now, imagine a new government program that decides to accept everyone because capitalisms is obviously flawed and full of greedy people.

The cost of such an insurance will be, for certain individuals, higher than what they would be with a targeted coverage. (The office workers for example.) Those individuals have now every incentives to leave this insurance and either take a private one, or even take none at all.

This causes the remaining few in the government insurance to have higher premiums, causing even more people to leave and check for alternatives.

The only way to prevent this exodus is to force everyone to stay with a single, monolithic insurance coverage. Even (especially!) those who are healthy.

Let's assume that you are OK with the concept, because without the government, the sick, single moms and children are facing certain death...

Because this super insurance is working outside of the market, you can't evaluate the value it provides. (If you force someone to buy an ice cream cone $36 at gunpoint, is it really worth $36? Maybe, maybe not...) This in itself is a huge problem, because you never know when you are wasting too much resource to try to solve a problem. Would this money better spent financing schools? Perhaps it would be better to distribute better food to poor families? Again, it's outside the market so you'll probably never know. While there's an infinity of bad/suboptimal ideas, there's only one or a handful good. Now pick one and 'hope'. Throw in a little political games in it, and you'll probably never end up with what you were expecting to begin it.

But there's yet another, more 'local' problem. Now that you have your too-big-to-fail insurance, how can you keep it in check? Should it cost $1k to process a claim, or $10k? When is your administration too big? You can compare yourself to some other insurance companies, but that's assuming you didn't wipe out the entire market (which you have to, eventually, to prevent the exodus). Even with comparison, you never have a decisive proof that it's too big. How do workers react, when you try to resize an industry, without the proof that it needs to be done? If, in addition, it's a government subsidized industry, good luck ever making it smaller.

The end result is a "service" that not a single soul knows how much it should cost, but with a constant pressure to increase the funding.

Also, this is without taking into consideration some pharmaceutical firms' lobbying.
Imagine how happy they would be, if none of their consumers were the payers!
(It's already a problem in the U.S. IMO; people use drugs but just forward the bill to their insurance.)

These problems only grow with time. While a government insurance might be "not so bad" in the beginning, every structures and forces around will deteriorate it. And before you know it, you won't ever be able to talk against any of its part, because then you'll be a greedy capitalist that wants to leave the poor and sick die a painful death.

This is the trap I'm talking about. When you're in it, there's no way out.