Since officers these days usually don't have their own horses and so the military provids one, it would likely be unusual to ever see another color used as it is simpler to simply procure a black one and dispense with the additional covering gear of tradtion.
But in cases where the deceased did have a favorite horse, and had expressed a desire to use it, it may not always be black. Hence the hood and cape mentioned above. I've never seen it done except in old records but evidently it's permissable.
And see the above article concerning a horse Ron received from Austria. Reagan's Lippizzaner came up during a FR thread on Patton which is why I recall it; it wasn't as well known as the arab he was known to ride.
I'd want a mare rather than a stallion, I've always gotten along better with the girls. I've had the privilege of riding a Conversano Lipizzaner mare that belonged to an Austrian lady - she was marvelous, but very forward and strong, you always had the impression you were sitting on a case of dynamite.
I can imagine the Secret Service would want to veto Reagan riding anything, especially if they aren't horse people. And they'd pitch a fit over a stallion just on general principles. Ever since Henry VIII (yet another thing he's famous for), the English-speaking countries have gelded most male horses, so riding stallions is rare and many folks simply are afraid to because of the folklore that's grown up around them. But as long as you have some Vicks VapORub to put in his nose when the ladies are in season, and assuming that he doesn't have temperamental problems, stallions are generally o.k. I still prefer mares though, and they seem to prefer me.
The black horses in British funerals, at least in pre-Victorian times (Victoria forbade the use of black horses at her funeral), wore the full caparison even if they were black underneath. They also had black plumes on their heads.
I still have something like a caparison in my tack trunk. It's an open macrame-like full cover with tassels at the end where the cord is tied off. Only it's white cotton and it's now called a cooling sheet. I guess in an emergency I could dye it black . . .