If it's your barn you could do what I did.
Found cheap shelves at IKEA and hung them.
My "grooming boxes" are actually baskets I found on sale that fit on the shelves and hold all the brushes, etc I need to groom.
*friend of bar.ka*RIP all my lovely boys, gone too soon: Steppin' Out 1988-2004 Hey Vern! 1982-2009, Cash's Bay Threat 1994-2009
Sam(Jaybee Altair) 1994-2015
A french cleat is the way to go. Zero protruding parts on either box or wall, and quite secure. Well, it would be susceptible to being knocked upwards, such as a horse deciding to rub its head on it, but I can totally picture an easy fix for that.
You can make one from wood, which would look nice, or just buy a picture hanger, available at any of the big box hardware stores.