Marcy, when the phone company says it blocks calls without caller ID, they mean without a number, the name doesn't apply here.  The reason it says "out of area" or "unknown" or some such is because your phone company couldn't look up the name for whatever reason.  But if there is a number at all, it's "valid" caller ID even without a name, which is one of the major failings of the blocking services.

The 000-000-0000 number is probably how your phone company is identifying people that called you with a blocked number (no caller ID), but chose to record their names for you to listen to when you answer the call and decide whether to take the call or not.  This is the only way someone without caller ID on their line (calling you from a workplace, for example) can get through to you.  This is how it works for me with Qwest/CenturyLink anyway.  So you probably don't want to block 000 numbers.  Check your phone company's documentation.