You invite a homeless relative in Florida to live with you on a temporary basis and after five months ask them to leave. They insist that they have gain property rights because now it is their resident address and you will have to have them evicted. They have been living there free for the five months paying for nothing. Have they gained property rights in the property simply by the fact they lived there for five months?

Yes if somebody stays overnight one night and they receive mail at that address you have to evict them.
Give them a three day notice and then if they do not leave go to the magistrate and file a formal notice.

Guidelines for Florida Residential Eviction (http://rhol.com/csu/evictions/Florida/FloridaSteps.htm)