Rights are creations of human law. So animals have whatever rights we legislate for them. So we should instead ask: should we give animals any rights, and if so, which ones? The answer is yes, we should legislate at least some rights for animals, insofar as compassion compels us to protect animals from unnecessary suffering. But animals aren't people. There is therefore no warrant for treating them the same as humans, and thus we should extend to them fewer rights.
Scientifically, we know animals don't have the same array of thoughts, feelings, desires, or perceptions that we do, they can't enter into social agreements, and are unaware of their own existence and have no comprehension of death or any ability to ruminate on their experiences. Primates, cetaceans, elephants, and a few species of bird come closest to humans on these measures, and thus warrant more protections, but all animals fit onto a wide spectrum, from barely any appreciable mental life (worms) to near-human sentience (orangutans), and everything in between. Our legal and moral regard for them must be attenuated along the same scale.
The question whether we should eat them "if we wouldn't want them to eat us" evokes an example of the difference between animals and people: first, no animal is capable of entering into such an agreement, hence we could never avoid being eaten by a bear or a shark or a mountain lion by refraining from eating them: the principles of reciprocal treatment won't apply, because we cannot enter into any moral or legal arrangement of reciprocity with an animal; second, no animal is aware of what we are aware of or capable of what we are capable of, and any reciprocity of regard must account for these differences. For example, if I were a shark or a cow I would have no self-awareness, no moral character, no cognitive knowledge, no ambitions, no fear of death or even a cognitive love of life, all I would know are sensations and animal emotions, and I would have no better prospect than a life of mindless labor and a miserable death anyway, even if I lived entirely in the wild. I would thus not mind if I were eaten, so long as I were not beaten.
As I would not mind being eaten if I were a shark or a cow, I do not mind eating a shark or a cow. But as I would mind being beaten if I were a shark or a cow, I do mind beating a shark or a cow, and would legislate against it if I could.