VA Hiring Enhancement Act This bill modifies requirements and procedures related to the hiring of physicians at the Department of Veterans Affairs (VA). Specifically, the bill negates enforcement of a noncompete agreement that an applicant for a physician position in the Veterans Health Administration (VHA) has entered into with a non-VA facility or entity. As a condition of terminating the noncompete agreement, the physician must agree to serve at the VHA for one year or until the expiration of the noncompete agreement, whichever is later. The bill also authorizes the VA to offer a contingent physician's appointment to a person if the person will have completed a residency program leading to board eligibility in a specialty and met other requirements for the appointment within two years after the VA offer. Eligibility requirements for a VA physician's appointment are revised to require completion of a residency program rather than an internship.