It's not really the 10th amendment that is going to govern this issue. This issue will be governed under the commerce clause of the constitution. This means Congress will get to do it. Right now, legal scholars have a great debate going on concerning the commerce clause. The SCOTUS has defined it so broadly that just about anything that Congress wants to do it can in the name of interstate commerce. Healthcare is something that affects commerce across the states. That means Congress has a very broad discretion to meddle into it. This legacy comes down from the reinterpretation of both the general welfare clause and the interstate commerce clause that occurred in the same era as New Deal legislation.

In Modern times, the 10th amendment is (normally) only violated when Congress attempts to commandeer a legislative or other process of the states. An example is the old laws where: Congress provides that each state must arrange for toxic waste disposal of waste generated within its borders. Congress requires the state to "take title" of the waste if it fails to comply and thus become liable for tort damages stemming from it. (New York v. United States) Congress can't compel state legislatures to enforce federal policy under the 10th amendment.

Another example is administrative offices: Congress can not compel a state's sheriff's to perform background checks on applicants for handgun permits. (Printz v. US) Congress can make it illegal and enforce it with federal agencies, but they can not force a state agency to do anything specifically to enforce federal policy.

As for the commerce clause, generally, Congress may enact laws that cover four broad categories:

1. The Channels of interstate commerce: This covers the regulation of highways, waterways, and air traffic.
2. The instrumentalities of commerce: This refers to people and machines (trains and semi trucks) used in carrying out commerce.
3. Articles moving in interstate commerce: The goods themselves crossing state lines etc.

And finally the big catch all that gives them so much power:

4. Anything "substantially affecting" commerce: So long as the activity is "arguably commercial" then it doesn't matter if the particular activity itself directly affects interstate commerce so long as it is part of a general class of activities that, collectively, substantially affect interstate commerce.

Medical insurance falls into category 4. You can't buy insurance over state lines. That means that most "particular instances" of insurance are not interstate commerce. However, taken collectively, insurance has a substantial affect on commerce when you look at it countrywide. Now, you can see all kinds of examples where this will make Congress have an almost unlimited right to legislate. So many things, when taken in the aggregate, fit this definition. Legal scholars are still wondering what exactly can Congress -not- do? Only a few recent cases have put any real limits on it. It's sad, but we are now seriously waiting to not find out if Congress -can- do a thing.... We're asking "is this one of the rare instances where they -can't- do it?"

It's messed up, but that's the current state of constitutional law.

The Constitution specifies, among other protected rights, that we cannot be slaves - prohibiting not just the government from owning slaves.

That's actually incorrect. There is an amendment prohibiting a "person" from owning another "person" and we also fought a war over this premise. However, it is elgal for the government to own a person. Just ask any of the many military personnel in the US who were courts martialed and/or received nonjudicial punishment for "willful destruction of government property" when their reckless behavior resulted in an injury. These cases are for as little as getting a severe sunburn from falling asleep on the beach that results in missing work. Is it against the spirit of the constitution? Absolutely.. Is it the current state of the law? Absolutely not. It always amazes me how many people don't realize that the government can own people. I'm just glad no corporation has stretched the rational to try to own a person. The governments argument is that the war and the amendment forbid "people" not the government from ownership.