Where do gun rights come from anyway?

If the Civic rights model says that the Second Amendment only addresses the Congressional power to Arm the militia granted under Article I, Section 8, Clause 16, is there such a thing as gun rights under that interpretation of the Constitution?

Yes, there is, but it doesn't arise from the Second Amendment. It comes from State Constitutions.

I was putting off this post, but Fat White Man made the comment:

Ohio has it right in our state preemption law:

9.68 Right to bear arms - challenge to law.
(A) The individual right to keep and bear arms, being a fundamental individual right that predates the United States Constitution and Ohio Constitution, and being a constitutionally protected right in every part of Ohio, the general assembly finds the need to provide uniform laws throughout the state regulating the ownership, possession, purchase, other acquisition, transport, storage, carrying, sale, or other transfer of firearms, their components, and their ammunition.

As I said in response, he's on the right track here by citing state law.

Under the Constitutional system, The Bill of Rights is a baseline. States can guarantee broader rights than that granted by the US Constitution. States cannot offer less protection than that guaranteed by the Bill of Rights. Thus, we have the Second Amendment, which is narrowly interpreted for the right to apply to the Militia:

A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.

One of the legal maxims I keep mentioning is Expressio unius est exclusio alterius (The express mention of one thing excludes all others) : Items not on the list are assumed not to be covered by the statute. That is, you can't infer that a statute, or other legal writing, applies to something if it is not specifically mentioned somewhere in that writing. (So keep looking in the Constitution for "self-Defence").

On the other hand, state provisions explicitly allow for non-militia uses. Examples:

Colorado: The right of no person to keep and bear arms in defense of his home, person and property, or in aid of the civil power when thereto legally summoned, shall be called in question; but nothing herein contained shall be construed to justify the practice of carrying concealed weapons. Art. II, § 13 (enacted 1876, art. II, § 13).[Self-defense right explicitly protected.]

Connecticut: Every citizen has a right to bear arms in defense of himself and the state. Art. I, § 15 (enacted 1818, art. I, § 17). The original 1818 text came from the Mississippi Constitution of 1817.[Self-defense right explicitly protected.]

Delaware: A person has the right to keep and bear arms for the defense of self, family, home and State, and for hunting and recreational use. Art. I, § 20 (enacted 1987).[Self-defense right explicitly protected.]

Georgia: The right of the people to keep and bear arms shall not be infringed, but the General Assembly shall have power to prescribe the manner in which arms may be borne. Art. I, § 1, ¶ VIII (enacted 1877, art. I, § XXII).
1865: "A well-regulated militia, being necessary to the security of a free State, the right of the people to keep and bear arms shall not be infringed." Art. I, § 4.
1868: "A well-regulated militia being necessary to the security of a free people, the right of the people to keep and bear arms shall not be infringed; but the general assembly shall have power to prescribe by law the manner in which arms may be borne." Art. I, § 14.[Self-defense right protected, McCoy v. State, 157 Ga. 767 (1924).]

Idaho: The people have the right to keep and bear arms, which right shall not be abridged; but this provision shall not prevent the passage of laws to govern the carrying of weapons concealed on the person nor prevent passage of legislation providing minimum sentences for crimes committed while in possession of a firearm, nor prevent the passage of legislation providing penalties for the possession of firearms by a convicted felon, nor prevent the passage of any legislation punishing the use of a firearm. No law shall impose licensure, registration or special taxation on the ownership or possession of firearms or ammunition. Nor shall any law permit the confiscation of firearms, except those actually used in the commission of a felony. Art. I, § 11 (enacted 1978).
1889: "The people have the right to bear arms for their security and defense; but the Legislature shall regulate the exercise of this right by law." Art. I, § 11.

Illinois: Subject only to the police power, the right of the individual citizen to keep and bear arms shall not be infringed. Art. I, § 22 (enacted 1970).[Self-defense right protected, Kalodimos v. Village of Morton Grove, 470 N.E.2d 266, 273 (Ill. 1984).]

Indiana: The people shall have a right to bear arms, for the defense of themselves and the State. Art. I, § 32 (enacted 1851, art. I, § 32).
1816: That the people have a right to bear arms for the defense of themselves and the State, and that the military shall be kept in strict subordination to the civil power. Art. I, § 20. [Self-defense right protected, Kellogg v. City of Gary, 562 N.E.2d 685, 694 (Ind. 1990).]

Louisiana: The right of each citizen to keep and bear arms shall not be abridged, but this provision shall not prevent the passage of laws to prohibit the carrying of weapons concealed on the person. Art. I, § 11 (enacted 1974).
1879: "A well regulated militia being necessary to the security of a free State, the right of the people to keep and bear arms shall not be abridged. This shall not prevent the passage of laws to punish those who carry weapons concealed." Art. 3.

Get the idea?

State provisions explicitly mention personal uses of firearms, but don't confuse the Second Amendment with State Constitutional provisions.

We have this quote showing a difference between use for the common defence and personal uses from ratification times:

The Bill of Rights secures to the people the use of arms in common defense; so that, if it be an alienable right, one use of arms is secured to the people against any law of the legislature. The other purposes for which they might have been used in a state of nature, being a natural right, and not surrendered by the constitution, the people still enjoy, and [may?] continue to do so till the legislature shall think fit to interdict. “Scribble Scrabble,” Cumberland Gazette, January 26, 1787; “Scribble-Scrabble,” ibid., December 8, 1786

The right to keep and carry firearms was one of the issues in Commonwealth v. Selfridge (1806), the most important murder trial of the early republic that began to change the need to retreat in self-defence situations. In this case, Selfridge’s lawyer conceded that “every man has a right to possess military arms” and “to furnish his rooms with them.” Yet the defense also recognized that the ownership and the use of non-military weapons were not constitutionally protected. Rather than assert a constitutional claim, the defense framed a common law argument on behalf of his client. Selfridge’s attorney argued “there is no law written or unwritten, no part of the statute or common law of our country which denies to a man the right of possessing or wearing any kind of arms.” Given this fact, it was indisputable that “in every free society a man is at liberty to do that which the law does not interdict, nor can the doing that which is not forbidden be imputed as a crime.” Therefore, the acquittal in the Selfridge case made perfect legal sense since Selfridge had not broken any law.
Trial of Thomas O. Selfridge, attorney at law, before the Hon. Isaac Parker, Esquire, for killing Charles Austin, on the public exchange, in Boston, August 4, 1806 by Thomas O. Selfridge, Published by Russell and Cutler, Belcher and Armstrong, and Oliver and Munroe (Boston) 1807.

Likewise, there are other instances where it is made clear that States legislatures could regulate firearms for personal uses:

The legislature, therefore, have a right to prohibit the wearing or keeping weapons dangerous to the peace and safety of the citizens, and which are not usual in civilized warfare, or would not contribute to the common defense. The right to keep and bear arms for the common defence is a great political right. It respects the citizens on the one hand and the rulers on the other. And although this right must be inviolably preserved, yet, it does not follow that the legislature is prohibited altogether from passing laws regulating the manner in which these arms may be employed.

To hold that the legislature could pass no law upon this subject, by which to preserve the public peace, and protect our citizens from the terror, which a wanton and unusual exhibition of arms might produce, or their lives from being endangered by desperadoes with concealed arms, would be to pervert a great political right to the worst of purposes, and to make it a social evil, of infinitely a greater extent to society, than would result from abandoning the right itself. Aymette v. State, 21 Tenn. (2 Hump.) 154 (1840).

Aymette is quite clear on the difference between the use of arms for Common defence and personal uses.

But don't look to the Second Amendment for your "gun rights" since that was one of the Tenth Amendment rights which was granted to the States.

So, McDonald v. Chicago gets it backwards by saying that the Second Amendment right should be incorporated to the States--It's the other way around.

State Constitutional provisions grant gun rights, not the other way around.

I should add that Article VI, Clause ii States:

This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.

That means The US Constitution is controlling law over state provisions.

Thus State Constitutions can explicitly provide greater "gun rights", but it is ultimately the US constitution prevails.