The big problem with speaking of natural rights in this way is that people are suggesting, by appealing to logic, that there are non-empirical things that are obviously part of us, that compel action. This is doubtful for several reasons. I'll treat the question begging premise of being "natural" first, and then the issue of rights second.
Natural things
First, because you can't posit normative facts by an appeal to logic (well you can in theory, but not here), as logic simply tells you how to describe the world exactly as it presents itself, through reflecting statements. Logic of course only talks about inferences that cannot not be the case. Natural rights, I suggest could well not be the case - it's possible to imagine this present world without them. It is therefore not the job of logic to posit them. I have a hard time seeing how it is that natural rights could be something the world would include as part of its internal structure, as the world probably has no idea we're here.
Second, this view of natural rights is suggesting a prescriptive quality is found in descriptive things - implying an obviousness in how the world presents itself for response (well, the world possibly has no idea we're here) e.g. if a person is drowning, there is nothing more that is obvious aboutthe situation than that they are drowning - anything else is a matter of personal reflection, not logical compulsion. In other words, even if somebody believed they had "natural rights", it says nothing more than that they believe they have natural rights. It says nothing about the world that I have to believe or act on.
Finally, a descriptive fact has no ability to compel an action e.g. that the sky is usually a shade of blue, therefore we must worship the god Woden doesn't follow. The sky is blue therefore it is blue. Nothing else. How we fit these things around our ordinary lives is another matter of course, but it is not a matter of logic, necessity or obviousness, and certainly not a precription of nature. If certain rights were prescribed by nature, we'd have no choice but to act upon them. A day in many African countries easily proves that it's possible to be otherwise. Therefore, "natural" is a bad adjective here. Logic therefore requires nothing more than for us to say only what we have no choice but to say. All else is interpretation, and we'll all agree that interpretation is useful but by no means such that have to see a thing that way.
Rights
Now I'll treat the question of rights themselves. A right is something you claim from somebody else, not something you're born with. This is the case for reasons shown above, and more precisely, because of an incosistency that results otherwise. A right is an legitimate expectation, or a given permission. Where a right is a legitimate expectation, it is expected because you have a reason to claim it.
Reasons as rights
Private rights
A reason is an evidential claim - it tells that, or why something is true. If what is true is a matter of logic, then, as explained above, what is true demands a very strict criteria of verifiability. As such a legitimate expectation here is where you have a reason for expecting a liberty or an action. A promise, for example, creates a right to expect fulfilment of the other side. A promise, being a verbal agreement, creates in the memory of the speaker, a piece of evidence of intention of the other side, and thus an expectation of a particular action, and thus a right.
Public rights
A legitimate expectation can, however, be created when an authority makes a promise or sets a limit to its competence. Permission, thus creates rights. The government's legislation puts into the public access, reasons for justifying certain freedoms they claim from eachother and from the government. This is what political and legal rights are. I cannot therefore agree that a political right needs the same origins as other rights
Moral rights
But I think that it is moral rights that are disturbing us most here. Morality, many will agree, is not all that clear to those who haven't the time that philosophers have to think about it in ideal settings. Therefore, I will stipulatively define morality as those considerations we take when measuring the human quality of a situation. Which says nothing more than how satsified I am from the outcome of actions that others have taken against me. Why can I expect not to be killed for my property at will? Well, because the person who does it to me would not have it done to them. They would, if I put them in my shoes, employ the same plea that I would. And it is this that creates what we'd think of as moral rights. However, moral rights are not propositions - they don't say things about the world, but about us, and so once more we have the problem I described in the opening. A proposition is an exact statement about a fact - it it cannot ot be true. So to say "murder is wrong" is not a proposition - as it doesn't say anything that's important to the definition of murder - it isn't a piece of the picture that I can add whenever I see it missing somewhere. In other words, there is no evidence I can use to claim that murder is wrong, except that I don't like it. But so what if I don't like it? It still says nothing that's descriptive of murder. There is therefore no "right" not to murder. having showed what it is not possible to say about moral rights, I will in the end say how itis possible to speak of them.
Rights without reason
One of my objections to natural rights in the Enlightenment way is that it speaks nonsense. It purports to say what it couldn't possibly say. If we take a right to be what I described above, it seems to be that they are descriptive of a situation or relationship. They're a piece of the situation. How can self-accredited things relate to anything other than the self? My name is Gilliwoo, it's a property of me. I have two legs, they're a property of me. I own a phone, it's not a property of me.
Moreover, how can statements about what I'm feeling inside, be descriptive of the world? When I say I have a right to an education, without a political or moral support for it, I am expecting something without any evidence that I have reason to expect it. I just feel it should be. This may therefore be nothing more than pompous godwottery about what I reflected on today. I'm simply describing my thoughts. But how does something that describes, make it that you should do things in response to it? They don't without a receiving ear. Because until somebody could threaten your right, you have nothing to claim exactly. As such it requires at least a mutual relationship (and this is one of my justifications for society as a valid concept). Natural rights are logically impossible things. They suggest that an action can be compelled before there is anybody to compel it upon. If you're born with something, it should be there if you were the only person in the universe. A right, being a claim, cannot exist without at least two people. Otherwise it's just a conversation with oneself.
A right not threatened is just a description of intent, and that has no substantive worth. It has no value because it has no scarcity. But something that I claim from somebody else (e.g. a natural right of free speech) needs justification. Appeals to "nature" don't justify anything at all - nature likely doesn't even know you're using it as evidence! And the other guy has no reason to believe you anyway, and rightly so, because the only authority justifying that right is you, which means I couldn't take it away from you if it became a burden on me. I won't let you have it, therefore, your right just once more becomes a declaration, not a reason. Speaking of a natural right is jumping out of your own skin. It's nowhere written that this emotion requires that response. Unless God put them there and told us what they were therefore, it's hard to even know where to start (even if God exists, the question is not settled whether God telling you things could make them true).
How morality remains possible
I constantly appeal to moral requirements in discussion here. How am I able to do this if I deny natural rights? Because I'm not tied to your mast. Moral rights are open ended. That is to say, while I can never say "I have a right to an education - end of", I can say "You've not yet given a good reason why I should not expect an education, and I've plenty of explanations why I should". In other words, moral claims are possible because they don't claim facts about the world, but facts about a situation. Situations are composed of physical objects, but we conceive them as abstractions. The way we think about the situation is thus more important in deciding how to behave, than what is actually there. e.g. a person plunging a knife into an innocent person is only that. But we receive this situation with various other pieces of knowlegde such as our knowledge of pain, our fear of death, our hatred of injustice. The moral quality of the situation is bigger than the sum of its parts. And because we understand situations abstractly - in relation to how we happen to feel and think and nothing more perfect - there is no logical reason to exclude these sensibilities in our reasoning about them. In other words, I flip the logic of natural rights: rather than saying there are logically necessary properties in the world that create rights, I say that there are no such things at all, but there is no reason not to include what we'd like to happen, in how a situation operates. It is THIS that creates claims, and implicit or explicit consensus that upgrades them to rights. When I claim to have a right to be heard fairly I'm claiming that there's no reason why I shouldn't be. When I'm saying that politics should include moral considerations, I'm claiming that there's no reason why it should be assumed that my feelings about the tax levied on me are somehow irrelevant to the situation - I live with it therefore it is something that has real meaning to me. It isn't, however, a necessary reason to care what I think about taxes. In other words, the fact that there's no reason not to consider facts with feelings, if feelings will be part of the end result, makes it impossible for logic to forbid me from considering those things (this is a bold statement, and I suspect Phawkins and Drunkhamster may have issue with it!)
Conclusion
Some people may have lost patience with y long post by now, and wonder why I must "intellectualise" the issue. I don't believe I've done this. i've only shown that you can't think the things you know you believe, and believe this also. I have shown fairly reasonably, that if we accept the actual "axioms" of existence - logic and the world around us - we are forced, to downgrade "natural rights" to things that we care about, and are part of the situation just as much as the facts are. But that they are equivalent to facts, is not true. They don't describe anything, they're just how we want things to be. And that is no bad thing to consider.