If you're working at their direction using their equipment and their location for work, you're not a contractor no matter what the boss says. That's just a warning in case they are trying to get around the rules that bigger companies have been fined for trying.

You don't need a tax ID or anything. Your best bet is to set up a separate "business" checking account that's not really a business checking account (since the banks like to charge you more for that). Send invoices regularly, deposit the checks in the new account. Withdraw amounts on a regular basis to pay yourself for the work. Keep your books in something like Quicken or Quickbooks or Money just to make sure you're following where the money comes and goes. Then when the new year comes around, look for a good accountant to get you through tax season (and, yes, you'll owe a bunch, but the accountant can help with that as well).