No, because natural rights come from free will. Animals, as far as we know, are deterministic organisms, unlike humans.
Free will grants us an axiomatic, praxeological basis. We know that humans act. To deny this, you would have to act, proving yourself wrong. As a corollary to human action, we know that humans have free will. Without free will, human action would not be purposeful or meaningful. In order to deny this, you would have to use your free will to deny the existence of free will, therefore proving yourself wrong. From free will, we know that there must be an originator of free will - what we shall call the mind. The mind is the originator of all thoughts of a given being, making it the legitimate owner of it. So because of the existence of the mind, we know that every being with free will is a self-owner. From this, we can deduce that all property homesteaded by such beings is owned by the homesteader, since this property ownership is a natural extension of self-ownership.
From all of the above, we can deduce that anyone who coerces or uses violence on another being capable of free will essentially enters a logical contradiction: by denying the free will/self-ownership of a being capable of free will/self-ownership, the action-originator denies his own free will/self-ownership. The action-originator implicitly allows any other being capable of free will/self-ownership to exact justice for the victim against the action-originator by using force.
So there we have it. A value-free system of justice that establishes certain "rights."