There's a very fundamental difference between the US mindset and the European mindset, which I think is often overlooked when we try to understand their attitude to things like nationalised health care and gun control.
In the very broadest of terms the European mindset, even when a Conservative party is in control, is roughly socialist. That's socialist as in "for the greater good of all", so we often accept higher taxation and more state interference in return for an overall fairer society, although this can be at the expense of the individual.
The US mindset is, in contrast, roughly libertarian. They cherish the rights and freedoms of the individual over society as a whole, so any one who attempts to limit individual freedoms, such as the right to own guns, or suggest that people should pay more tax to support a not for profit medical system are seen by many as anti American.
Of course the above boils something huge and complex down to its absolute basics, and there are plenty of Europeans who are more libertarian leaning and Americans who are socialist, but in very general terms it explains a lot about why America is the way it is.