EDIT: The employee was obviously doing something negligent (stupid) by not having the horse tied and separated from other horses whilst removing the blanket. Unless the barn owner specifically told him/her to remove a blanket under those conditions, thereby placing the employee in harm's way, the employee can't sue-- and should have known better.
If you're such an idiot that you stick your face into a paper shredder in an office, you can't sue your boss either.
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Most states have laws that basically say, since activities with horses (or animals in general) are inherently prone to risk, you can't sue a horse owner or stable owner unless an injury is received that is due to a lack of normal safety precautions or unreasonable risks. So getting randomly hurt by a horse in a freak accident, through no fault of anyone, when the horse has no record of hurting people, usually isn't actionable...
...but there are exceptions. Different states are very loose or very harsh about these laws. Here's a link where cases in each state, related to horse injuries, have been decided:
http://asci.uvm.edu/equine/law/cases/laws/laws.htm
The other poster is also correct; if the employee wants money, he should be talking to the barn owner, not the horse owner. The barn owner can then maybe go after the horse owner for repayment. The horse owner has no responsibility for the safe working conditions at the barn owner's barn.