Social Security Number Privacy and Protection Act of 2004

Social Security Number Privacy and Protection Act of 2004 - Directs the Director of Selective Service to alter the form of the Selective Service reminder mailback card, or the method by which the card is submitted to the Selective Service System, to reduce the risk of theft of Social Security account numbers included as part of the identifying information required from persons presenting themselves for registration under the Military Selective Service Act.

Amends title XVIII (Medicare) of the Social Security Act (SSA) to eliminate the Social Security account number from Medicare, Medicaid (SSA title XIX), and SCHIP (SSA title XXI (State Children's Health Insurance) identification cards.

Amends Federal veterans' benefits law to eliminate the Social Security account number from veterans health care identification cards issued by the Department of Veterans Affairs.

Expresses the sense of Congress that health insurers should not use Social Security account numbers on insurance identification or claims cards issued to beneficiaries, but should substitute another identification code or number instead.