Health savings accounts are theoretically flawed, even if globally applied and available to all in the way that their advocates envision. They proceed out of individual savings and affect or help not at all those who can't save. More generally, they abstract money from the system which is thereby less able to provide care for the poor and unemployed.
The positive social good of insurance is a broadened risk pool's capacity to absorb catastrophic risk. This requires a risk pool composed of those less likely, as well as more likely, to need help. The profit imperative, on the other hand, demands a risk pool restricted to those less likely to file claims, and as much parsimony and delay as possible in meeting claims. Those are in conflict, and, in my view, fatally so for those looking to private insurers for solutions to health care problems. Those advocating health savings accounts play into the companies' goal of restricting risk pools, rather than a public interest in broadening them. They might be good for low-risk individuals who can save their otherwise higher insurance premiums. Not so more generally.
Another example of placing individual priorities above those of the larger polity--selfishness, some might call it--in service of the false and morally questionable notion that We're virtuous and deserving, and Their misfortunes, being Their Fault, have nothing to do with us, and We shouldn't help Them. Demonstrably false, even pragmatically, much less as a belief system which might underlie a better world, or better policy.