You make several valid points; let me address them one at a time.
> ... first off, you used the communal computer and I told you not to. ;-)
Well, it's not really communal if it can't be used for the enlightenment of others, is it? ;-P
> If the concept of rights is not valid, then NOBODY can justify or not justify use of property. Therefore, you cannot justify the use of "your" computer.
Agreed. But this situation is not self-contradictory. However, this situation is different from the situation in which anyone can justify any use of any property, which is my understanding of the communal situation. The communal situation is self-contradictory. The proof by contradiction of the original poster fails, I think, because there is no logical transition from the first situation to the second.
> Rights are a concept. They do not exist in the physical world. We cannot physically touch a right, but if we are to live in a civil world then we will have to come to agreements.
Agreed, but this does not imply that rights are subjective. It merely implies that our conception of rights is subjective. In the same sense that our conception of truth is subjective, but truth itself is objective. Natural law, like truth, is objective. By the same token, rights (being the extension of natural law) and truth do not exist in the physical world but in the world of ideas.
> If rights are not valid, then you do not have any moral argument to defend your speech.
No, but there may be other arguments by which to defend speech. "I get pleasure from arguing" is a valid argument which does not rely on natural rights or any moral argument.
It is not a valid argument, of course, if in your definition of "defend" you incorporate a moral component. Then no defense which does not boil down to a moral argument is a valid defense. But that'd be choosing the definitions so as to prove what you already agree with, rather than changing your opinion to agree with whatever can be proven from logical deduction.
> If a person thinks rights are not valid, then they should not complain if someone punches them, steals their property, tells them to shut up, or shoots them. And they should not complain when anyone does any of those things to anyone else.
Quite the contrary. If a person believes rights are not valid, then their complaining about being punched is entirely in accordance with their belief system as they need no justification to do so. However, their complaint less meaningful because the complaining-behavior is in accordance with their belief system even when they're not being punched.
It must be noted, however, that a valid complaint can still be made on the basis of empathy and pain while making no appeal to rights.
> If you try to claim that rights do not exist, then you will have to use your body in some way to do so, which proves that you believe that you exercise a right to speak, write, or type. By trying to disprove rights, per se, you are demonstrating that rights do exist. It is axiomatic for that reason.
While I agree with the approach of argumentation ethics ("since you are arguing, you cannot argue against rights"), I find this particular argument to be lacking. I can write a simple program that outputs the string "I think that rights do not exist." when when queried with "What do you think about natural law?". Moreover, suppose I am someone's slave and that person has ordered me to argue in favor or in opposition of rights. In this case, by arguing, I am not asserting my own property rights in my own body. On the contrary: I am affirming my owner's property rights in me by doing as he says.
The ability to argue and the act of argumentation are no more proofs that rights are valid than the ability to steal and the theft itself are proofs that the thief is the rightful owner of the recently acquired property.