Sounds like garden variety extortion to me. Start with your local municipal government office and police. They may be able to shed a lot of light on this. This contractor is committing a crime, and you're probably not the first victim. Be sure to find out if this contractor required any kind of license to perform this work (unfortunately, many state and local governments specifically waived a lot of contractor licensing requirements that are normally in effect). Find out of this contractor is licensed ANYWHERE. If so, file a complaint with that licensing authority, and with the Better Business Bureau in that area.
Where is the written contract with your signature, where you agreed to pay this amount for certain specified work? Doesn't exist, I'm sure. Call your county courthouse and put in writing to whoever they tell you, that you have reason to believe that someone may attempt to file a fraudulent lien on your house. That might be enough to stop the lien from being filed, if the court demands more information when (if) they attempt to file.
Since the threat refers to specific federal agencies, send a copy to your Congressperson, with a letter of explanation. I suspect this construction outfit is violating some federal laws by sending this letter, even without actually following through and filing the lien.
Contact your local newspaper and TV station. They may not do anything right away, but if they get multiple reports of the same type of thing, either by the same contractor, or by multiple contractors, they'll start making a lot of noise, and that will light a fire under local authorities.
Be aggressive. Let the contractor know that you know that filing this false lien would be committing a crime, and that you will make sure they get nailed to the wall if they don't cease and desist immediately, and that you'll probably make sure they get nailed to the wall even if they DO cease and desist immediately. Let them know that a copy of their letter, and accompanying letter of explanation from you has been sent to the local, county, and state law enforcement, the Florida Department of Labor, your Congressperson, FEMA, the US Army Corp of Engineers, PRTS Inc., Direct Installation Services, Bramer Construction, your insurance company, and your mortgageholder. And really send it to all of those parties. Scare them. Let them know they made a big mistake in targeting you. I'm sure they're doing this to multiple people, and they don't really care if they have to back away from a few. Put yourself at the top of their list of targets to back away from.
No, they cannot take your house. They are hoping to scare you into "settling" for a smaller amount, but still more than you or anyone owes them. But if they actually file a lien (which they may not have any real intention of doing) they can cause you all sorts of hassle and expense to get it lifted.
To everyone who replied - thank you so much for your help. I'm really going to kick some butt here!!