Visa Overstay Enforcement Act of 2017 This bill amends the Immigration and Nationality Act to subject any alien who remains in the United States after any visa or status under which the alien is lawfully present has expired to: (1) a fine or imprisonment for up to six months, or both, for a first offense; and (2) a fine or imprisonment for up to two years, or both, for any subsequent offense. Such penalties shall be waived if the overstay was due to medical necessity, public safety, or national security. An alien convicted of a first offense may not be admitted to the United States for 5 years or granted a visa for 10 years. An alien convicted of a subsequent offense may not be admitted to the United States at all or granted a visa.