I don't see any evidence of these "rights" occurring in nature - they are human constructs born of our aspirations and expectations.
You're definitely correct. As I mentioned in another post, the framers of the Declaration believed the rights they stated to be endowed by our creator, and were intrinsically qualities of "human" life. Since these are naturally "human" constructs, who is to say that they are wrong? Other thinkers throughout time have had similar trains of thought: "I think, therefore, I am" being another one. For reasons that others have put more articulately than I could summarize, humans have always believed themselves to be "special" from all other life on earth.
government can and does deprive people of liberty constantly. And even, in some cases, their lives.
Yes, but life and liberty are not guarantees - just that you are born with these rights, and that the government cannot take them away from you when you are born. The government can take away your life and liberty, but - as you pointed out - through due process as a result of criminal activity. If your crime takes away, deprives, or limits these same rights of others, then the government will no longer offer its protection of those rights for you - and under certain circumstances, can take them away.
It seem more useful to define a right as "that which it is morally unthinkable to deny".
My problem with this is that there is going to be a very big gulf of opinions about what is moral and what isn't. As you say, the right to a healthy life seems pretty reasonable to you ... but I would reason that its actually unreasonable to expect to always be in good health. In the case of healthcare, there is also a great divide about what constitutes reasonable care. Many governments that have socialized medicine also do offer insurance for those that can afford it so that they can get better care. Why would those insurance companies exist if the care the government provides (or pays for) is sufficient? If the answer is that some people do not think it's good enough and want better care, wouldn't that also, by your own definition, mean that the government is not satisfying the rights of it's people, and thus would be a failed state? I have no problem with socialized healthcare, or a welfare safety net ... but I would consider these programs to care for our least well off, and not "rights".