TL;DNR: As is so often the case, we can't give a definite answer to your questoin. In many states, the owner could extort the neighbors by installing an spite fence, but these are illegal in Florida. And under Florida law, it is also possible the neighbors have a legal right to use the land, but the requirements are so strict it seems unlikely they can. We need to know more before we can say for sure what the legal outcome will be.
Eye Sores, Spite Fences & Spite Houses: There is a long history of Americans using what the OP calls "eyesores" to extort their neighbors who refuse to buy their land. Eye sores are so common, they have their own name -- spite fence -- Wikipedia page, and laws. (In Seattle, there is even a spite house!) Most states control spite fences by imposing restrictions on the height of fences. In Florida, case law says a fence is a nuisance if it was built only out of malice.
Adverse Possession: The most obvious justification for the neighbor’s use of the property is adverse possession. Adverse possession allows someone who has been using another’s property for a long time without permission to get title to it. Under Florida law, adverse possessors must also have either: 1) Paid the property taxes regularly; 2) Have relied on a filed title. Since this property was just sold at tax auction, the neighbors obviously have not been paying the taxes. As for the title requirement, who knows?
Easements It is possible the neighbors could have an easement to use the property. Since the article doesn’t mention an easement, it seems unlikely they have an explicit one. Under the common law of Florida, prescriptive easements (easements by possession) cannot be given for the entire property. Whether this use is “limited” enough to justify a prescriptive easement is not clear.