So how does this apply to particles? Does this statement [point to] wave particle duality?
No, the fact of Snell's law cannot be thought of as being any evidence for the wave particle duality of light, even though Frobenius' Answer / Feynman's derivation superficially considers a particle's path.
This is because rays of light can equally well be interpreted wholly in wave terms, namely, as the unit normals to the phase fronts of waves. Whenever solutions of the D'Alembert / Helmholtz wave equation fulfill a slowly varying envelope approximation, the Eikonal equation follows and Snell's law for ray normals at interfaces is the inescapable conclusion of the Eikonal equation. In turn, all these concepts are equivalent to Fermat's "least time" principle.
The slowly varying envelope approximation is basically that, over regions of less than several wavelength's diameter, the wave can locally be thought of as a plane wave with a well defined phase front, i.e. that the solution $\psi(\mathbf{r})$ of the Helmoltz equation as a function of position $\mathbf{r}$ is of the form $\psi(\mathbf{r}) = \Psi(\mathbf{r})\,\exp(i\,\varphi(\mathbf{r}))$, where the amplitude $\Psi(\mathbf{r})$ is real-valued and varies significantly only over regions much larger than a wavelength. Over regions of a few wavelengths the phase is well approximated by $\varphi(\mathbf{r})\approx\mathbf{k}\cdot\mathbf{r}$.
A ray is then the integral curve of the vector field $\nabla\varphi$, and the more slowly the amplitude $\Psi(\mathbf{r})$ varies in comparison to a wavelength, the more accurately the Eikonal equation and Snell's law hold.
I show how to derive the Eikonal equation, Fermat's principle and the slowly varying envelope approximation from each other in this answer here and the answers that one links to.
But there is also an experimental answer that can be given to your question. Waves in wave tanks moving across interfaces between regions of different, constant depth can be experimentally shown to fulfill Snell's law. The Eikonal equation and Snell's law are also widely, successfully used in seismology and other, wholly wave-governed fields of physics.
