Let's start off with the founders were very familiar with the work of Thomas Hobbes. Hobbes reasoned that this world of chaos created by unlimited rights was highly undesirable, since it would cause human life to be "solitary, poor, nasty, brutish, and short". As such, if humans wish to live peacefully they must give up most of their natural rights and create moral obligations in order to establish political and civil society. This is one of the earliest formulations of the theory of government known as the social contract.
In short, too many rights can create chaos. The founders did not want anarchy (or what they would call "democracy"). In fact, they saw republican systems as being somewhat more restrained, although the French revolution would prove that to be a fallacious belief.
On the other hand, one of the things tied in with natural rights is a right to life, which unlike the current one which ends at birth, this one BEGINS with birth. Natural rights begin with existence. The Declaration of Independence says that: "We hold these truths to be self-evident, that all men
are created equal; that they are endowed by their Creator with certain
inalienable rights; that among these are life, liberty, and the pursuit
John Locke stated the right to life as being that everyone has a right to life once created. The fundamental law of nature is
that human life, as much as possible, should be preserved. We could have a longer discussion of Locke, but his ideas are not supportive of a belief that natural law or natural rights would allow for the ownership of weapons. In fact, I would assert they seriously contradict that belief.
Contemporary statements at the time of the US's nascence show that the doctrine of self-defence required that force was used as a last resort and only allowed for as much force as was reasonably necessary to stop any threat. If one exceeded that level of force, then the person who was attacked became the aggressor 
The Constitution provides the right not to "be deprived of life, liberty, or property, without due process of law". While one can argue that acts of "self-defence" are not governmental actions, the way that the government addresses the use of that doctrine ARE.
The real issue when one claims a right, is what authority backs up that assertion? In the case of life, there are many international treaties that say killing without extraordinary justification is wrong. Article 3 of the Universal Declaration of Human Rights, which the US is a signatory, states "Everyone has the right to life, liberty and security of person."
Recently, the United Nations human rights committees overseeing the United States' obligations under the International Covenant on Civil and Political Rights (ICCPR) as well as the International Convention to End all forms of Racial Discrimination (ICERD) meeting in Geneva Switzerland both cited rampant gun violence in the United States as a violation of our government’s duty to effectively protect the right to life. Women, children and minorities were cited as disproportionately affected groups. Both treaties have been ratified by the U.S. Senate binding the US government to act.
Both committees recommended expanded background checks covering all private firearm transfers. Both also expressed concern about the proliferation of Stand Your Ground laws stating that these laws are used to “circumvent the limits of legitimate self-defense in violation of [the U.S. government's] duty to protect life”, and requested review of those laws “to remove far-reaching immunity and ensure strict adherence to the principles of necessity and proportionality when deadly force is used for self-defense.”
Albert Camus said "bloodthirsty laws, make bloodthirsty customs." While he was speaking specifically about state-sponsored executions, the tendency goes like this: once the citizens of the state allow their government to discipline its citizens by killing them, then killing citizens becomes a custom; the punishment becomes, then, customary, in Camus' sense of the word. The laws rise from the consent of the governed, and once in place, they start to do their job: determine the culture of the country. The state's custom then becomes a little more bloodthirsty because the state's laws allow us to slake our thirst for blood.
The real issue here is that rather than "gun rights", any proper appeal to natural law or fundamental law would more likely hold that the right to life is far more important than someone's need to own a gun. In fact, US law in recent times has perverted many concepts held in civilised nations regarding self-defence, right to life, and even the rule of law.
 Blackstone’s Commentaries on the Laws
of England, Book the Third, Chapter the First: Of the Redress of Private
Wrongs by the Mere Act of Parties p.3