Male friend has been living in a female friends home for approx 18 months. Male friend is alcoholic and behavior has become disruptive to the household. This friend contends that because he has lived here 18 months he can't be forced to leave. There was only a very loose verbal agreement of rent. What rights does the female homeowner have in this situation?

She has the right to evict him formally, under the local laws.

Give him formal written notice that they want him to move.
The amount of time to give them will depend on state law ( if in the US) if they don't move out within that time frame you file for an eviction in court.