As this is an area that is new and developing, there may be no hard and fast rules as yet. The law lags technology, usually by a wide margin, and it is the early cases that actually help to make the case law that subsequently guides all who follow.
First, it may depend, at least somewhat, on who your employer is and how reasonable the request to sign the new policy is deemed. Second, it may depend on which computer is used to access such social media sites (as you state, if you never 'go there' on the work machine, how can they ask for your personal password information), and third, it may have something to do with how public your presence on line actually is (check your privacy settings).
In the end, always remember that Florida is an "at will" employment jurisdiction meaning that a private employer can fire an employee for almost any reason, or even for no reason at all. Since the law in not exactly clear in regard to social media interactions, what must occur to make it clear is for some people to be fired and then for them to file suit so the issue can come before a court which would then interpret the law and come up with the first decisions in the area.
So, to answer your question, you could be ordered to provide them with passwords and you could be fired for refusing, particularly if you sign on to this new policy. Is that right? Well, maybe not - the courts would have to decide. Best bet is to keep your accounts private, never use a work machine for personal business, and avoid signing anything if you can. Good luck.
This is a very controversial and interesting area of the law. Increasingly courts and legislatures are recognizing that individuals should have rights of privacy in their social media and social networking activities. By the same token, however, employers are increasingly concerned that the activities of their employees in social networking activities can harm the reputation and business of the employers.
It certainly is well established in most states in the United States that if you use your employer's computer facilities to engage in these activities, you give up your right to privacy. Note that this remains controversial--in Europe the opposite is true. And many are arguing that employers in the U.S. (and court decisions in their favor) have gone to far in stripping employees of privacy rights.
Far more problematic is for an employer to assert that it has the right to have access to your social networking activites using your own computer facilities, cell telephones, I Pads, etc, My best guess is that courts and legislatures will eventually prohibit employers from insisting on the right to obtain access to private social networking activities.
More fundamentally--some have argued that there is nothing left of the right to privacy in teh United States. Everything we do leaves a footprint somewhere, whether on a computer hard drive, cell telephone text message record, GPS system recording your location and whearabouts, voice message, or other such internet based communications. The increasingly conservative courts, federal and state, have not been kind to privacy rights. The Fourth Amendment protections against unreasonable searches and seizures have been drastically limited in recent court decisions, and the balance has shifted decidedly in favor of law enforcement. Concerns about terrorism have resulting in a fundamental shift in the way public officials and judges think about our fundamental privacy rights and liberties. I think most people would be truly alarmed at the extent to which they have lost their privacy rights. But in recent years, efforts to reverse this trend by progressives have not gone very far--the Patriot Act keeps getting renewed every year, and court decisions continue to eviscerate previously well-established Fourth Amendment rights. Courts have held that you essentially lose all privacy rights when you drive an automobile, and police can set up random roadblocks to stop and check on whether drivers are impaired by alcohol or drugs. Courts also have held that your privacy rights are severely limited in airports, hotels, public places such as stadiums and theatres, and essentially everywhere other than your bedroom. In short, the Fourth Amendment protections against unreasonable searches and seizures (which incorporate rights of privacy) don't provide much protection these days---which is truly a shame. It will take a profound political movement if this trend is to be reversed.