A "right" can only exist if there is some means of implementing it.
Therefore, for the purposes of this discussion, asking whether healthcare is a right amounts to exactly the same thing as asking whether there should be the means to offer healthcare for everyone.
That could be attained by sufficient civilian iniative, but in practice it is a function of what is perceived as the duty of government towards the citizens - and how it is codified into laws.
Whether access to free or affordable healthcare should be implemented as a (legal) right is a considerably separate matter. And a a far more interesting one, IMO.
We already have the right to health care, since access to health care is not denied to any citizen, based on the law.
Consider the second amendment which is the right to bear arms or own guns. The individual has the right to own guns, but the government does not provide free guns to everyone who wants one. The right only gives one the option to possess at a gun or seek health care, but a right is not a freebie program. A freebie program is called an entitlement. One is not entitled to have a gun, but one has the right to own one, but they need to jump hoops and find a way to provide for themselves. A right assumes an adult is in the room.
Progressives tend to confuse rights with entitlements. An entitlement was something that the King and Queen had, while rights is something the peasants eventually got. Rights restricted the entitlements of the King and Queen, so the peasants had more options. However, the peasants had to provide for themselves, since the King and Queen were entitled to everything else. The King and Queen owned the army which was needed for self defense. The peasants got the right to bear arms so they could defend themselves if the King and Queen needed the army for other things.
The government is entitled to health care since they have your tax dollars, make the laws. and run the police and justice departments. However, they cannot take all the best doctors in the country for themselves, since the peasants have the right to health care. But they need to figure a way to pay for it; private sector.