Hoppe's argument in favor of self-ownership is that in the process of arguing a person asserts his property rights over his body and maybe some other objects; therefore, if he argues that he does not have these rights, then he is engaging in a performative contradiction. He is exercising the rights even as he claims that he does not have them.
First things first. In one sense, argumentation is an action. We are trying to get from an unsatisfactory state, namely, not knowing the truth, to a preferred state, namely, knowing the truth, and we are using a means: argumentation. In another sense, however, argumentation is the opposite of action. Action involves use of the body and tools and natural resources, though, of course, motivated by a desire (in the will) and facilitated by a plan of action (in the practical intellect). But argumentation is a purely intellectual, speculative activity. It presupposes autonomy and self-control but not a body.
In other words, if argumentation could be conducted telepathically, as, say, between angels, so much the better; in fact, the less argumentation depends on physical makeshifts (the brain, the speech organs, the air, etc.), the greater your autonomy and the better for you in your capacity as arguer. Angels and God have superior ways of knowing (and therefore, I'd imagine, arguing) precisely because of their incorporeality.
The distinction between the active life and the contemplative life is time-honored and reasonable.
Now we are not angels. But with respect of arguing, we wish we were. In other words, bodily functions and physical events such as propagation of sound waves through the air are quite accidental to arguing, while they are not accidental to human action. Human action in the second sense essentially involves pushing particles of matter around.
So, Murphy and Callahan are right that, at best, Hoppe's argument succeeds in showing that control (not ownership) over certain parts of one's body (not the whole body and not other property) is necessary in order for a certain activity (a scholarly debate) to go on among those participating in this activity (not all people) during that time (not at any other time).
E.g., Hoppe's argument fails to establish the content of my property rights. Exactly what am I allowed to do with my property and what are you allowed to do so as not to infringe on my rights? How is the "bundle of rights" to be distributed? The argument is sterile. At most, it shows that you are allowed to use your property for arguing, nothing more.
Also, the argument proves too much: if air is necessary for debate, must air be owned? Clearly, one can be a libertarian without holding that air must be privatized. So, in the process of arguing one asserts control not ownership over air which suggests that he need only to assert control not ownership over all the other resources.
Now there are two ways of resolving the contradiction. First, the person can acknowledge that he does have the right to his body and stop arguing against it. Second, he can acknowledge (perhaps in his mind) that he does not have such a right and also stop arguing against it. So, the same action, namely, stopping making a particular argument, can be a sign of two completely different conclusions.
Two important things do not follow from "I can't argue against my own having these rights":
(1) That I have these rights. Being quiet is not an argument for self-ownership. How does my failure to be able to argue over a point represent concession of the point? I may still be unconvinced by your arguments, even if I can't offer anything in response. The end result may very well be "I don't know if I have any rights."
For example, even if correct, Hoppe's argument establishes that non-libertarianism with respect to self- and property ownership is self-defeating -- it comes without ground for believing in it, because, according to Hoppe, any attempt to establish such non-libertarianism results in a performative contradiction; it does not establish that it is self-refuting -- it may still be true, even if we can have no reason to believe it.
(2) That you have these rights. I can agree that I have a right myself, but I need not extend it to others. Hoppe would have to make an additional assumption that I am so interested in the truth about this subject that I am willing to countenance the rights of other people to their speech organs and suchlike in order to receive feedback to my argument.
Citing the idea that ethical rules must be universalizable at this juncture would be a red herring. The point is that I may think that I (a) have / (b) don't have the relevant rights and argue against your having rights without contradiction.
In (b) I may be using my bodily, etc. powers unjustifiably (or not: perhaps I've been permitted to use them by their true owners), but that is at most a moral failing not an intellectual one (and that may be true of performative contradictions in general).
In (a) there may be some relevant difference between you and me, such as: you are a criminal.
There is, however, a way to salvage the argumentation ethics by pointing out something like the following. In order to engage in a philosophical debate it is arguable that the debaters must be surrounded by a high civilization. People who are barely surviving have neither the time nor the inclination to discuss philosophy. But high civilization is impossible without property rights and freedom and peace. So, any upstart philosopher who wants to do his thing must acknowledge his debt to liberty and thank his stars that he lives in a free society. It has been said that capitalism has, by increasing enormously the standard of living of everyone, created a class of "intellectuals" whose main occupation is denouncing capitalism. Now that is a performative contradiction if there ever was one.