Actually wanting to die - as opposed to just executing a behavior which leads to your death - requires that you have the concept of your own death. Is there any evidence that any other species even has that concept?

That seems pretty obvious to me - animals are not stupid - but I don't really know what type of evidence you would accept.

http://www.inquisitr.com/44905/amazing-photo-of-a-chimpanzee-funeral/

Anyway the premise seems wrong. By "wanting to die" all I meant was that the organsm engages in behaviour that leads to death. I wasn't suggesting spiders and praying mantis exhibited very much abstract thought. A plant can "want to die" - in that sense of the word. However, it is non-biological idea - something we don't expect to see much - and in fact don't see much.