First: understand your rights, and their limits. This is a common case in employment law. The general standard is that the employer must make "reasonable accommodation".
EEOC explains in more detail: https://www.eeoc.gov/laws/guidance/what-you-should-know-workplace-religious-accommodation
How does an employer determine if a religious accommodation imposes more than a minimal burden on operation of the business (or an "undue hardship")?
Examples of burdens on business that are more than minimal (or an "undue hardship") include: violating a seniority system; causing a lack of necessary staffing; jeopardizing security or health; or costing the employer more than a minimal amount.
If a schedule change would impose an undue hardship, the employer must allow co-workers to voluntarily substitute or swap shifts to accommodate the employee's religious belief or practice. If an employee cannot be accommodated in his current position, transfer to a vacant position may be possible.
Infrequent payment of overtime to employees who substitute shifts is not considered an undue hardship. Customer preference or co-worker disgruntlement does not justify denying a religious accommodation.
It is advisable for employers to make a case-by-case determination of any requested religious accommodations, and to train managers accordingly.
IANAL, but I would expect that they would try to argue that "our company does not work that way" is equivalent to a seniority system. Note that the canonical case on seniority systems vs Sabbath observance (Trans World Airlines, Inc. v. Hardison) involved a seniority system codified into a Collective Bargaining Agreement. Even then, the employer has a duty to try to make things work, see eg https://casetext.com/case/cassell-v-skywest-inc
Next, try to reach an agreement that can work for you and your employer. Suggest that you are open to any of the possibilities mentioned in https://www.eeoc.gov/laws/guidance/questions-and-answers-religious-discrimination-workplace, and in particular see if you can find someone willing to swap with you.
An employer may be able to reasonably accommodate an employee by allowing flexible arrival and departure times, floating or optional holidays, flexible work breaks, use of lunch time in exchange for early departure, staggered work hours, and other means to enable an employee to make up time lost due to the observance of religious practices.
Moreover, although it would pose an undue hardship to require employees involuntarily to substitute for one another or swap shifts, the reasonable accommodation requirement can often be satisfied without undue hardship where a volunteer with substantially similar qualifications is available to cover, either for a single absence or for an extended period of time. The employer’s obligation is to make a good faith effort to allow voluntary substitutions and shift swaps, and not to discourage employees from substituting for one another or trading shifts to accommodate a religious conflict. However, if the employer is on notice that the employee’s religious beliefs preclude him not only from working on his Sabbath but also from inducing others to do so, reasonable accommodation requires more than merely permitting the employee to swap, absent undue hardship.
Finally, consider calling the EEOC or a lawyer to see if they can help.
From the page above:
What should an applicant or employee do if he believes he has experienced religious discrimination?
Employees or job applicants should attempt to address concerns with the alleged offender and, if that does not work, report any unfair or harassing treatment to the company. They should keep records documenting what they experienced or witnessed, as well as other witness names, telephone numbers, and addresses. Employees may file a charge with the EEOC, and are legally protected from being punished for reporting or opposing job discrimination or for participating in an EEOC investigation. Charges against private sector and local and state government employers may be filed in person, by mail, or by telephone by contacting the nearest EEOC office. If there is no EEOC office in the immediate area, call toll free 1-800-669-4000 or 1-800-669-6820 (TTY) for more information. Federal sector employees and applicants should contact the EEO office of the agency responsible for the alleged discrimination to initiate EEO counseling. For more details, see How to File a Charge of Employment Discrimination, https://www.eeoc.gov/employees/charge.cfm.
One more thing, since eg @TheDemonLord seems to be unaware: the Court's conception of genuine religious belief tends to be quite broad, and dependent only on the individual's subjective understanding, rather than any official approval by others, including their own religious community or leaders. See guidance and cases cited at https://www.eeoc.gov/laws/guidance/section-12-religious-discrimination#h_9593682596821610748647076
Update: The Supreme Court has in fact decided the case of Groff vs DeJoy, and expanded the protections given to religious observance. In a 9-0 decision, the Court ruled that an "employer must show that the burden of granting an accommodation would result in substantial increased costs in relation to the conduct of its particular business." It is not immediately clear to me how this would apply in the context of a seniority system, but it certainly should give any employer pause before refusing accommodation.
See https://www.scotusblog.com/2023/06/justices-rule-in-favor-of-evangelical-christian-postal-worker/
