Veterans Health Care Eligibility Expansion and Enhancement Act of 2013 - Modifies provisions relating to health care services for veterans to: (1) require (currently, authorize) the Secretary of the Veterans Administration (VA) to furnish health care services to veterans who do not qualify under existing VA programs for veterans with a service-connected disability or other impairment; (2) require the enrollment in VA health care programs by December 31, 2014, of veterans who do not have a service-connected disability rating; (3) extend the period of the eligibility of veterans with certain combat service to enroll in VA health care programs; (4) limit requirements for providing hospital care, nursing home care, extended care services, and prescription medications to veterans based upon the amount provided in advance in appropriation Acts; and (5) revise criteria for treating veterans as low-income families based upon location and income factors for purposes of enrollment in VA health care programs.

Directs the VA Secretary to: (1) use the VA capitation-based resource allocation model (provides for the allocation of health care services based on factors such as population, patient age and sex, and financial need) in entering into contracts for furnishing health care services; (2) afford priority for entry into contracts for Federally Qualified Health Centers and Community Health Centers; and (3) modify guidance for VA health care services contracts to incorporate best practices in such contracts.