Honestly, I think your farrier is well intentioned but I would call a vet. Your farrier probably knows a lot but medical advice is best left to those who attended medical school.
I don't understand that if the tendon is lax (loose) than how is listerine going to tighten it? I ask this simply because a tendon is to my knowledge a lot like a rubber band. Once stretched beyond its capacity it is likely to either break or not contract as usual. The listerine might reduce swelling but it would not tighten anything. The way it would "tighten" gums would be to eliminate bacteria that would create plaque and bacteria in the mouth thus allowing the gums to tighten themselves because they are "healthier". It would do nothing to the tendons and ligaments, largely because the only tendon or ligament in the mouth is the periodontal one that holds teeth in the mouth. If you were looking for a product to reduce swelling and thus "tighten" things than I would look at an equine product. I would also look at hock boots for before or after rides. Those would support the tendon during the stress. Are you supposed to wrap the listerine hocks?
If your horse is having hock troubles than I would look into figuring out exactly what those troubles originate from is it a tendon or is it a joint/arthritis issue.