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The use of the word "socialism" is not limited to textbook economic socialism.  Words evolve.
Republicans complain that Social Security and Medicare are socialism.  They complain that universal child care is socialism.  They complain that public libraries are socialism.  
See?
Why not everyone?! We get sick too! And we’re poorer!! >

Wouldn’t that be socialism though? >

Socislism = workers own the means of production. 
So no, healthcare for all is not SoCiAliSm