The word here is a lien. It all depends on your state and county. Where I am from, if the individual must not have paid for three consecutive months. In order to gain ownership of the horse, you must first contact the owners giving them proper notification of the current circumstances. If they still do not come forward you must advertise in a local paper as having a sherifs sale of the horse(s) in question. The horse can only be sold for as much as is owed to you in board. Again, this is just from where I am form, chances are it is going to differ. My best advice to you would be to contact your sherifs department and let them know the situation. They will give you proper guidance where to go from there.
We had a similar situtiation at my parents home. A friend needed a place to keep her horse and we told them only a couple of months... We figured 60 - 90 days. Well after not seeing the girl feed the horse we tried calling her and her parents to get them to care for the horse. They would come and go but never really kept up with feeding the horse on a regular basis.
8 months later, a bucket load of feed and several round bales of hay. We gave them a bill and 30 days to pay up. The girl finally showed up and gave the horse to my dad to sell.
It really depends on your local laws. It also depends what the boarding contract sais. A few years ago I was escorted by the police to a barn near where I live to pick the horse of a boarder that had not paid board for 6+months. The horse was seized off the property.
I do not know what the details were on this one as I my job was to pick up the horse but I do know they worked diligently with a lawyer and local law enforcement.
I suggest looking at the contract signed then going thru a lawyer.