Accoring to [H. Lenstra][1], 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
$x^2+ny^2$*.

  [1]: http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf