That seems a strange argument for an individualist to make.
How so?
What if a new species of ape were to be discovered whose members were on average less intelligent than humans but more than other apes, say where there was a significant overlap between the intelligence of members of that species, and of our own? Or what about a human-chimp hybrid? Or a (natural or artificial) mutant chimp? Would any of them have rights?
I do not have an good answer to give you regarding chimeras. Regarding the hypothetical species, we may as well be arguing the attributes of an intelligent race of Flying Spaghetti Monsters. Such things are not known to exist.
I think that's all that anybody is arguing for. The obligations on us not to mistreat animals are essentially equivalent to a right of those animals not to be mistreated.
The way we erect the ethical framework regarding how we treat animals will color our future decisions. To give animals the same degree of moral regard that we give humans by granting them "rights" will increase human suffering.
Perhaps in the future we will use a term other than human rights.
I certainly hope not, absent the presence of another intelligent species than our own, perhaps from outside the Solar System. To do so otherwise is extensio ad absurdum.
BTS