I manage a small Bed and Breakfast. We have long term renters/tenanats or guests because of gas and oil, pipeliners in area. We do not allow cats, yet the room is being trashed and not kept cleaned. We gave verbal notification when they asked about cat we said NO Cats. We have restrictions and guidelines for dogs and they have not even followed those they are endangering dogs that may have to stay because they are service/therapy dogs. Can we evict on notice because of policies not being followed. We gave a verbal but they have not removed cat or cleaned room.
This is a question that can only adequately be answered by an attorney with knowledge of the laws in the state in which you reside. Having considerable experience at writing and interpreting laws, however, I'll provide some insights that may be helpful.
The first matter that stands out is your statement that you "gave verbal notification" regarding your Bed and Breakfast's policies with regard to pets. While the law does accept the concept of a verbal agreement, you would have been in a far better position had you presented all prospective guests/tenants with a written agreement they would be required to sign as a condition of residing in your inn. Had the tenants in question signed a lease or renters' agreement that stipulated that no cats were allowed, you would have been in a far stronger legal position.
Verbal agreements can be considered legally binding, but they also lend themselves to "he said/she said" disagreements that may never be resolved to your satisfaction.
With regard to the tenants' failure to maintain an adequate level of cleanliness, once again, you would have been in far stronger position had you required a signed lease agreement prior to accepting tenants into your establishment. Without a very good idea of the actual condition of the unit in question, I cannot state with any accuracy whether a threat to public health is involved. You might want to consider asking the local law enforcement agency to visit the premises and offer an assessment.
The first step I would take is to provide the tenants with a written request to vacate the premises by a date certain. If there was a security deposit, you are probably in a position to retain it in perpetuity. And, in the future, require tenants to sign a written agreement regarding the terms under which you allow them to stay in your B&B.