Bernie very cleverly used this term “Medicare for All” during his campaign. It conveys very well the meaning of his proposal.
But in reality, healthcare by right for the “lower classes” exists outside of medicare too. As if you are a congress person, you have a right to publicly funded healthcare, same if you are a federal employee, same if you are in the military, same (in principle) if you are in prison, same if you are a destitute (but with children), same if you are a poor child, same if you are over 65. What we have is a nightmare of course (and public funds go to private hands), but as far as the principle of right to health goes, many categories have it - so it is hard to find the reason why the rest of us commoners shouldn’t.