This paper extends the high-order Flux Reconstruction (FR) approach to the treatment of non-linear diffusive fluxes on triangles. The FR approach for solving diffusion problems is reviewed on quadrilaterals and extended for triangles, allowing the treatment of mixed grids. In particular, this paper examines a subset of FR schemes, referred to as Vincent- Castonguay-Jameson-Huynh (VCJH) schemes, which are provably stable across all orders of accuracy for linear fluxes in first order systems. The correction fields of the VCJH schemes are shown to represent a family of lifting operators which are used to enforce inter-element continuity of the solution and the diffusive flux. For diffusion problems, the lifting operators of nodal DG schemes are shown to be a subset of this family. Finally, numerical results are used to show the effectiveness of VCJH schemes for a range of problems, including the model diffusion equation and the compressible Navier-Stokes equations. Optimal orders of accuracy are obtained on unstructured mixed meshes of triangular and quadrilateral elements.