Are rights a human construct? Yes, obviously. As are ethics and empathy and many other things humans value to some degree. You might see those constructs expressed in similar ways in some other animals, especially among the Great Apes, but they only truly reach their human form in humans.
Rights are a human construct in that they only matter between humans, or between humans and something humans want to treat in a human-like way.
Rights don't exist apart from sentient beings. They only exist within the brain, while still having consequences, with regard to interactions between those bearing the brains, in the physical world. The Universe doesn't have rights or respect rights otherwise.
A rock will never respect anything's "rights", nor will a mosquito. The rock has no consciousness or will (free or otherwise) and a mosquito just does what it must to survive long enough to reproduce-- it doesn't concern itself with anyone else.
Being a construct doesn't mean rights are imaginary. They are real-- at least when you are speaking of human interactions. Life doesn't turn out well if you don't respect the rights of others at least a little bit. If you didn't, you'd be worse than the worst psychopath, and you wouldn't survive long. You'd be everyone's enemy and everyone would be doing all they could to end you.
So, rights are a useful construct. And as long as I'm dealing with other humans (or creatures I want to treat humanely) I will respect rights and will expect mine to be respected by other humans as well.