What does it mean when says returns do not have an ssn for each dependent care provider are not eligible for electronic filing

The IRS requires a social security number for all child care providers to efile because if you do not have the number you must provide other information (see below).

You must identify all persons or organizations that provided care for your child, dependent, or spouse.  To identify the care provider, you must give the provider’s name, address, and taxpayer identification number (TIN).  

If the care provider information you give is incorrect or incomplete, your credit may not be allowed.  However, if you can show that you used due diligence in trying to supply the information, you can still claim the credit. 

If the provider refuses to give you the identifying information, you should report on Form 2441 whatever information you have (such as the name and address). Enter “See Attached Statement” in the columns calling for the information you don't have. Then attach a statement explaining that you requested the information from the care provider, but the provider didn't give you the information. Be sure to write your name and SSN on this statement. The statement will show that you used due diligence in trying to furnish the necessary information.

Publication 503