Customer Non-Discrimination Act This bill amends the Civil Rights Act of 1964 to include sex, sexual orientation, and gender identity among the prohibited categories of discrimination or segregation in places of public accommodation. It defines: "sex" to include a sex stereotype, sexual orientation or gender identity, and pregnancy, childbirth, or a related medical condition; "sexual orientation" as homosexuality, heterosexuality, or bisexuality; and "gender identity" as gender-related identity, appearance, mannerisms, or characteristics, regardless of the individual's designated sex at birth. The bill expands the categories of public accommodations to include places or establishments that provide: exhibitions, recreation, exercise, amusement, gatherings, or displays; goods, services, or programs, including a store, a shopping center, an online retailer or service provider, a salon, a bank, a gas station, a food bank, a service or care center, a shelter, a travel agency, a funeral parlor, or a health care, accounting, or legal service; or transportation services. "Establishment" shall not be construed to be limited to a physical facility or place. Protections against discrimination based on race, color, religion, sex, sexual orientation, gender identity, or national origin must include protections against discrimination based on: (1) an association with another person who is a member of such a protected class; or (2) a perception or belief, even if inaccurate, that an individual is a member of such a protected class. The bill prohibits the Religious Freedom Restoration Act of 1993 from providing a claim, defense, or basis for challenging such protections. The bill prohibits denial of an individual's access to a shared facility (including a restroom, a locker room, and a dressing room) that is in accordance with the individual's gender identity.