Accoring to H. Lenstra, Chebotarev's theorem holds both for Dirichlet and for natural density (but he doesn't give a reference in this document). Applying Chebotarev to the extension $H/\mathbb{Q}$ where $H$ is the Hilbert class field of $\mathbb{Q}(\sqrt{-n})$ gives the result you want. (At least for primitive discriminants; for non-primitive discriminants you need an appropriate generalization of the Hilbert class field).

Added in response to Will's comment There is always a suitable field. Let $-D$ be a primitive negative discriminant and let $d=-m^2D$ be a general discriminant. Let $H_n$ be the abelian extension of $K=\mathbb{Q}{\sqrt{-D}}$ where a non-ramified prime splits iff its is principal and generated by an element of $\mathbb{Z}+m\mathcal{O}_K$. Such a field $H_n$ is called a ring class field and exists by class field theory. It also is an extension of $K$ by a singular value of the $j$-function.

Then $G_m=\mathrm{Gal}(H_n/\mathbb{Q})$ is a generalized dihedral group. There is a correspondence between conjugacy classes in $G_m$ and pairs of equivalence classes of $ax^2\pm bxy+cy^2$ of discriminant $d$, such that an unramified prime has its Frobenius in a conjugacy class iff it it represented by the corresponding form. This is why we can apply Chebotarev.

Accoring to H. Lenstra, Chebotarev's theorem holds both for Dirichlet and for natural density (but he doesn't give a reference in this document). Applying Chebotarev to the extension $H/\mathbb{Q}$ where $H$ is the Hilbert class field of $\mathbb{Q}(\sqrt{-n})$ gives the result you want. (At least for primitive discriminants; for non-primitive discriminants you need an appropriate generalization of the Hilbert class field).

Added in response to Will's comment There is always a suitable field. Let $-D$ be a primitive negative discriminant and let $d=-m^2D$ be a general discriminant. Let $H_n$ be the abelian extension of $K=\mathbb{Q}{\sqrt{-D}}$ where a non-ramified prime splits iff its is principal and generated by an element of $\mathbb{Z}+m\mathcal{O}_K$. Such a field $H_n$ is called a ring class field and exists by class field theory. It also is an extension of $K$ by a singular value of the $j$-function.

Then $G_m=\mathrm{Gal}(H_n/\mathbb{Q})$ is a generalized dihedral group. There is a correspondence between conjugacy classes in $G_m$ and pairs of equivalence classes of $ax^2\pm bxy+cy^2$ of discriminant $d$, such that an unramified prime has its Frobenius in a conjugacy class iff it it represented by the corresponding form. This is why we can apply Chebotarev.

Even more added A good reference for ring class fields is Cox's book Primes of the Form $x^2+ny^2$.

Accoring to H. Lenstra, Chebotarev's theorem holds both for Dirichlet and for natural density (but he doesn't give a reference in this document). Applying Chebotarev to the extension $H/\mathbb{Q}$ where $H$ is the Hilbert class field of $\mathbb{Q}(\sqrt{-n})$ gives the result you want. (At least for primitive discriminants; for non-primitive discriminants youe need an appropriate generalization of the Hilbert class field).

Accoring to H. Lenstra, Chebotarev's theorem holds both for Dirichlet and for natural density (but he doesn't give a reference in this document). Applying Chebotarev to the extension $H/\mathbb{Q}$ where $H$ is the Hilbert class field of $\mathbb{Q}(\sqrt{-n})$ gives the result you want. (At least for primitive discriminants; for non-primitive discriminants you need an appropriate generalization of the Hilbert class field).

Added in response to Will's comment There is always a suitable field. Let $-D$ be a primitive negative discriminant and let $d=-m^2D$ be a general discriminant. Let $H_n$ be the abelian extension of $K=\mathbb{Q}{\sqrt{-D}}$ where a non-ramified prime splits iff its is principal and generated by an element of $\mathbb{Z}+m\mathcal{O}_K$. Such a field $H_n$ is called a ring class field and exists by class field theory. It also is an extension of $K$ by a singular value of the $j$-function.

Then $G_m=\mathrm{Gal}(H_n/\mathbb{Q})$ is a generalized dihedral group. There is a correspondence between conjugacy classes in $G_m$ and pairs of equivalence classes of $ax^2\pm bxy+cy^2$ of discriminant $d$, such that an unramified prime has its Frobenius in a conjugacy class iff it it represented by the corresponding form. This is why we can apply Chebotarev.