Yes animals have property rights, they own their own body just like you and i do, in general they don't use that body to turn anything else from nature into useful capital, in other words property, they don't produce much, except birds nests etc. perhaps.
They don't need to sign a contract to own anything, just like you and i don't need to sign a contract to own property, we sign a contract to make a promise, an exchange, not to own property, at most to make use of someone else's property.