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.)
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?"?