A contract is a two-way street. Both parties have burdens and benefits. If one party violates the terms and conditions of the contract that might (note the conditional) relieve the other of further performance.
In this instance if the barn is not providing the competant and professional care implied (even if not stated) in an equine boarding agreement then they should move. If the barn owner sues under the contract the defense is failure to provide that professional and competant care. Of course the boarder is obligaged to bring proof satisfactory to the court that the terms and conditions of the contract were violated. At a minimum, photos showing moldy hay, poor fencing, dangers in barn aisles, etc. will be essential to defense under the claim.
Indeed if a suit were filed by the barn owner I'd counter claim for the expenses of the move based upon the same evidence that justified the move. If the OPs state has any "consumer fraud statutes" (and many do) they might well be entitled to damages (often doubled or trebled), court costs, attorney's fees, etc.
If you're in a bad situation then leave. Document the reasons why. Safeguard this evidence. If a "party" gets started then you're ready to address any issues presented.
Note that the above does not constitute legal advice the OP should consult with an attorney in their jurisdiction.
Good luck to the OP in the matter.
Member, State Bar of Texas (Retired)