I don't believe there's a single command to do this, but we can still accomplish it expediently. The idea is that the distance to the nearest different cell equals the distance to the nearest location whose immediate neighborhood contains more than one cell type.
Well, this is not quite true, but it's close: you might want to add approximately one cell width to such a distance. If you can live with such an approximation, here's the workflow:
Compute a focal variety grid using as small a neighborhood as possible, which would be a 2 by 2 square neighborhood.
Nullify all locations with a focal variety of 1 or less: these are the "inside" cells.
Compute the Euclidean distance grid to all remaining cells: these are the distances to locations along boundaries.
(Optional) Add the cellsize to the distance grid.
Extract the values at any points you choose.
To illustrate, here is a small piece of a grid colored by cell type. The black areas are the cells with focal varieties of 2 or greater (computed at step 2):
A hillshaded Euclidean distance grid looks like this:
It reaches peaks (maximum values) within the centers of each patch of the original raster and decays to zero along the patch boundaries, still shown with black.