Medary.com

Wikipedia definition:

Socialism refers to various theories of economic organization advocating state, public or common worker (through cooperatives) ownership and administration of the means of production and distribution of goods, and a society characterized by equal access to resources for all individuals with an egalitarian method of compensation. Modern socialism originated in the late 18th-century intellectual and working class political movement that criticized the effects of industrialization and private ownership on society, however, socialism itself is not a political system; it is instead an economic system distinct from capitalism. Karl Marx and Friedrich Engels used the terms "socialism" and "communism" interchangeably, and posited that it would be achieved via class struggle and a proletarian revolution. Vladimir Lenin, perhaps influenced by Marx's ideas of "lower" and "upper" stages of socialism, later used the word "socialism" as a transitional stage between capitalism and communism.

So, in what way is government-run universal health care access NOT socialist?

Why do advocates of socialism react so strongly (i.e. equating the word to "racist") when their socialist schemes are identified for what they are? If government control of economic activity is a good thing, a thought which is at the foundation of socialism, then why do socialists run so fast from the word?

It's because socialism doesn't work. Everywhere it's tried, socialism degenerates into either an apathetic welfare state, like Europe, or metastasizes into a totalitarian dictatorship, like Nazi Germany, the USSR, countless third-world dictatorships, or at best modern-day China. Socialism destroys wealth, but worse--it destroys the human spirit. And, all too often, it kills millions of people.

Socialism does not work.