Yes, horses can eat blackberries, like any treat, in moderation. They are actually good for them; they contain antioxidants.
Horses are usually pretty careful and will not eat sticklers. However, if they are going for a treat or are not paying attention, they may get stuck and/or cut on the spines. If those spines break off and stay embedded in your horse's mouth, that could be problematic.
"I hear and I forget. I see and I remember. I do and I understand." - Confucius