I’d call health a desirable public good rather than a right.
You make a fair point: I’m coming down on the negative rights side of that argument.
Some positive rights, such as the right to police protection and the right to counsel, are just another way of saying the government should defend individuals’ negative rights. Others, like the right to housing and the right to an education, use the language of rights (which everyone supports) to advocate for the provision of a particular good (which some may dispute).
But saying healthcare isn’t a right is not the same as saying the state shouldn’t work to guarantee access to healthcare.