Why healthcare, and not those other things?
Back in the day (before political activists started changing the definition), a right was something you could do yourself if government left you alone. You had a right to life, liberty, and the pursuit of happiness (you know, that thing). Not happiness, but the pursuit of happiness. A right to free speech, freedom to practice the religion of your choice. These are things you can do yourself if left unmolested.
Socialists and communists came along and started calling things that you got from other people "rights." They wanted to guarantee happiness, not just the right to work for happiness. Your post, and my post listed these so-called "positive rights" (which required the work of others) as opposed to "negative rights" (traditional rights that didn't require free labor of others). Healthcare, housing, etc. were actually listed as rights in the USSR's version of a constitution. But, you needed the power of government to take those things from others and give them to you...which meant that the people who created those things were not motivated to do that well or to do it in abundance...which is why you had a right to housing in the USSR, and housing sucked...and a right to food, and empty grocery store shelves, and famines, etc.
But, can I really have a right to the fruits of another person's labor? Democrats have always thought so. Remember when southern landowner Democrats used to have that right, before the Civil War? But, it seems more appropriate that I only have a right to work for the means to buy that other person's labor.