No, I don't think nonpersons have an "I," because that requires a level of self-reflective awareness that's incompatible with their nature. Being or not being a "thou" is a characteristic of persons, which is why I said it's a category error.
Keep in mind, this is all in the context of why I don't find the article compelling. It's because I disagree with one of the fundamental premises of its proposed hypothetical situation, one so fundamental that the thought experiment is dull without it.
I'm not here to change your mind. I'm just explaining why I don't find the thought experiment interesting. Suspension of disbelief about some core thing you believe to be true about the world is fun for the sake of science fiction stories, but it's dull for the sake of philosophical thought experiments.
I don't intend to convince you or get convinced. To me, the answer depends on the purpose of the question.
I'm curious now instead why you'd grant rights to entities that don't have an "I". Isn't that pointless? Or is it a mistake on my part, i.e. maybe there's no "I" but there's still a "you"?
In the "you" case, do you think the thought experiment could be reframed as an attempt at answering the question "where is the you?"?
I think an "it" can have rights. I'll go back to the horse. It has a basic natural right not to be beaten because it's a creature that can experience pain, not because it's an "I."
The legal system (at least in the U.S.) seems to need to call something a "person" in order for it to have rights (e.g., corporations), but I don't think the legal system is the origin of rights.
"I" and "thou" go together as references to personhood, so I wouldn't distinguish between the two. (And I'm mostly using "thou" here instead of "you" to show I'm referring to the concept rather than you personally. The "I/Thou" philosophical concept also has precedent of using the term.)
In that case, the question is still valid: how would you convince the non-person "I" that the "I" is dead?
(I can see how the original "person I" might be dead after such a change, but it doesn't really make the premise invalid.)