Everybody, this is happening to a lot of people right now: every billing mistake ever resolved with Bell Atlantic, NYNEX, etc, every ID fraud case already straightened out, etc, etc way back from the 1990s is suddenly coming to life again after Verizon sold off these bogus uncollectable debts to ANFI.
You can find a lot of resources out on the web about how to approach these cases, but most of them will tell you to send a dispute letter to them by certified mail and ask them for reasonable documentation (billing history, bills, judgements, etc) and telling them that you do not owe this money. Make sure the certified mail is return receipt requested so that you have proof you did this.
Next, you need to file complaints with the FTC, your state's department of consumer affairs, and the Attorneys General of Illinois, your own state, and the state where ANFI now alleges you incured the debt.
If they try to sue you or put this on your credit report, go get an attorney who specializes in consumer protection laws and sue them.