Believe it or not, the U.S. Constitution does not specifically mention health. As for regulating healthcare, many have taken the "promote the general welfare" notion from the Constitution's Preamble as an indication that the federal government can pass laws pertaining to health care. This then ties in to Article I, Section 8, which says, in part,
Where in the U.S. Constitution are health and property mentioned?
The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States...[and] To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.
Those who oppose the federal government's efforts to regulate the healthcare industry cry foul, saying that the Preamble is meant to outline the Constitution's goals and are not a condensed version of the document itself. They also believe that those parts of Article I mentioned above are intended to define Congress's abilities to levy taxes.
As for property, the Constitution does not explicitly state that individuals have the right to own property. During the time period when the Constitution was penned, the right of free men to own property was not in dispute and was therefore assumed in the Constitution.
The assumption of the right to own personal property is clearest in the Fourth Amendment, which states,
The right of people to secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause. . .
This amendment is what keeps authorities from being able to come into your house and search through your stuff without a court-ordered warrant.