Here's a digest of conversations I often have with opponents of private property.

You don't own your body, you are your body!

There's no conflict between the two ideas. To varying degrees you are your body (you are your brain to a much greater degree than you are your little toe), while you are also the owner of your body. You have the exclusive right to determine how your body may be used. This is what ownership means, at least in the context of scarce rivalrous resources like human bodies.

Granting this, there’s no obvious difficulty with going further. If you own your brain, and also the lock of hair you cut from your head then perhaps you can be the owner of things physically more distant still. These owned things might include what some call the means of production.

Just because someone doesn’t want to get raped doesn’t mean they believe in “private property rights”.

I agree. I’m saying that people generally believe that they have a right to exclusively control how their body is used by others–not merely that they have preferences about how it may be used (they do have such preferences of course). That belief about a right to exclusive control is a belief in private ownership. When we talk about a right we’re talking about a claim that we believe the use of force is justified in enforcing (when enough people agree, this right will be reliably enforced).

If the idea of rights is a sticking point we can decompose the claim like this:

A person considers a thing his property if he believes he’s justified in using force to exclude others from its unauthorised use.

People consider themselves justified in excluding others from the unauthorised use of their bodies, so it's clear they consider their bodies to be their property.

I understand the impulse to oppose private property from a wish to improve the situation of the have-nots, and minimise the advantage of the haves. But I think this approach gets things backwards. The right to private property is one of the most important protections of the poor against the wealthy. Absent any legal protection of property rights, the advantage that the wealthier person has over the less wealthy one increases at least in the following sense: the wealthier of the two always has more means with which to take the poor person’s property by force. The institution of private property helps level the playing field to the advantage of the poor person.