No, they are merely a non paying renter. The biggest worry is not paying real estate taxes, where after a couple years the property will be sold for taxes.
The person living there, if they pay the taxes each year are just paying the taxes, gives them no right to home what so ever.
Now if they let it go unpaid, and the state sells the house for taxes, who ever bids the highest at the tax sell will own the house
Go to the county web site where the house is, many will have property taxes and the such online. You can see what years have been paid, what is due and so on.