The Third Reich's political economy was one of pervasive state control.
On health care, for example, Hitler believed in national health care; being Hitler, he saw it as a lever for state control and, ultimately, his goals of racial purity & eugenics.
Now, it's important to recall that the Nazi platform was much more socialistic than the Nazi regime once it had absolute power. Social welfare and worker power were the sales pitch, not so much the policy. The Nazis crushed unions, for example.
Of course, the Communists crushed unions, too. The use of pleasant-sounding socialist economic fantasies to sell what turns out to be stark tyranny is an enduring theme.
The Atlantic in 1932, on the mishmosh of statism, racism, right-wing nuttery, and primitive economics in early Nazism: theatlantic.com/magazine/archi…
My favorite part of that article is where the Atlantic writer knocks Hitler for being too much like Gandhi: 
Anyway, the role of socialism, as such, in Hitlerism varied over time and was - like much of Nazism - intellectually incoherent & paranoid. But the incontestable truth is that the Nazis believed in an all-powerful national government on whose favor citizens depended.
Thus, the conservative argument is not that socialism is Nazism. It's that the road to things like Nazism or Communism can be signposted with fantasies of Left or Right, but it can only be driven in the vehicle of big government.
