As stated, businesses are not subject to First Amendment and the U.S. (and State/Local) Government is not subject to the First Amendment when acting in capacity as an employer. The Civil Rights act does prohibit discrimination based on protected class, including religion.
U.S. Jurisprudence, the government can bar speech (religious or otherwise) or practice only under "Strict Scrutiny" which means that the government must have a compelling interest, must narrowly tailor the law to achieve the compelling interest, and must use the least restrictive means possible.
To demonstrate strict scrutiny, I like to use the quintessential Free Speech demonstation of Burning the U.S. Flag: A law on the books banning the burning of the U.S. Flag speciffically is unconstitutional. However, one can be arrested for burning the U.S. Flag under other violations of the law such as an ordinance about burning anything in public during conditions favoring wildfires (narrowly tailored in that the burning of the flag was not specifically banned and the city has a compelling interest in not burning down the town and is restricted only to when wildfire conditions are present) or arson (narrowly tailored in that it bans burning of things that are not yours, is least restrictive in that it doesn't ban you from buying a flag to burn, and the government has a compelling interest in stopping people from burning other people's things.).
Federal Laws that restrict potentially religious practices are unconstitutional if they do not allow for exceptions for religious practices without meeting strict scrutiny (it must meet all three conditions or it is not constitutional) (Gonzales v. O Centro Espírita Beneficente União do Vegetal). One example from historical laws is that during Prohibition, the laws had a special exception for Sacramental Wine and the Government today allows for religious exception in its drug laws. States, however, are a different matter as the government cannot compel strict scrutiny in matters beyond religious exemptions for prisons and land use (i.e. you cannot deny a church a building permit because a religious structure violates a zoning law . I.E. You cannot say no to a church that uses sacramental wine because it wants to build in an area zoned to prohibit distribution of alcohol to the public) (Employment Division, Department of Human Resources of Oregon v. Smith).
Civil Rights laws in the United States usually have a standard of "reasonable accommodation" when meeting the needs of protected classes. Usually this means that unless the place of employment is unreasonably burdened in their business, they should meet with a religion's customs. Employment law also typically applies to businesses that employ more then a fixed number of individuals. There are also exceptions where the job requires discrimination (i.e. If the script calls for a black woman to play the lead, a white man cannot sue the studio because his audition wasn't considered. Similarly a religious organization can refuse to hire an atheist or non-believer if discussing matters of faith is part of the job).
So if you were part of a religion that requires you to not work on Mondays, this can be reasonablely accomodated by most employers as there are other faiths that have similar restrictions (Of the Abrahamic faiths, this day changes between Friday (Islam), Saturday (Judism, Some Sects of Christianity), and Sunday (Christianity, some sects more than others), depending on your specific belief and sect). The only way to get around this is if the employer is specifically hiring to fill a Monday schedule vacancy. However stranger tenants may be required to leave at home (i.e. Ritual Animal Sacrifice is unreasonable in the work place, what with the cleaning of the blood and the feeding of the goat until the hour is of the sacrafice... to say nothing about how off putting it is to patrons of the coffee shop you're employed at who have options.). In the government, religious holidays are usually permitted in some capacity (flex hours and leave), unless you are in an essential position, which again, because it needs to be done in 24/7, it's okay for the government to not let you go home to family on Christmas Day (there usually is special bonus pay for working on these days.).
Again, when making the determination, the employer needs to show that the accommodation is unreasonable. In states, 32 states also follow strict scrutiny when applying laws even if they are not intended to burden religious belief, so it will apply there. Federal Employment laws will also follow this standard, as will any federal law, but the specific ability of the Federal Government to mandate it to states (save for in prisons and in zoning and land use laws) was struck down by SCOTUS though the law applies Federally.
Edit: The first amendment is considered one of the three "Safe Guards of Liberty" amendments, and the entirety of the bill of rights of the United States Constitution are considered both "negative rights" and "individual rights". This means that someone has a negative right if they have a right to not be subject to the actions of another. Thus, all rights in the first 10 amendments are rights of the individual citizen to not be subject to specific actions of government. Thus, the government may not make you meet any criteria to declare your belief valid and all legal suits on the first amendment start with the premise of the Speech (Religious or Otherwise) is Protected until proven otherwise. Thus, the Government cannot compel you to meet some requirement to prove you're a "real" religion. So if you make up a religion and sue your employer for not accommodating a belief of this religon, the default operation of the judge is to assume that yes, Jedism is a religion and that the employer must show that it isn't. This also holds true for the U.S. Government. A conscientious objector to war need not be objecting as a tennante of religion so long as the belief stems from moral or other problem with all wars (you can't get the status if you object to this specific war but not World War II or the Civil War. And even then, the status does not mean you cannot be drafted, just that you can't have a duty that requires you to man a weapon. There are certain roles needed by the military which prohibit armaments, such as medics or chaplains. In fact at least 3 recipients of the Congressional Medal of Honor, the highest military award in the U.S., were given to conscientious objectors working in non-combat roles. Of the three, two were awarded posthumously. Given the criteria is for one to "distinguished himself conspicuously by gallantry and intrepidity at the risk of his life above and beyond the call of duty" one must admit you definitely qualify if you did all that while refusing to take up arms.).