Medicare Universal Product Number Act of 1998 - Amends title XVIII (Medicare) of the Social Security Act, as amended by the Balanced Budget Act of 1997, to direct the Secretary of Health and Human Services to establish a system for assigning a universal product number (UPN) to every covered item of durable medical equipment, including surgical dressings and splints, prosthetic devices, braces, and artificial limbs. Requires that each request for payment, or bill submitted, after February 1, 2001, for a covered item include its proper UPN.