If she’s going to discriminate against which customers she serves, she should lose her job. If she doesn’t want to touch men who aren’t related to her then that’s her choice, but when this is incompatible with the duties of her job, she needs to decide which one is more important to her.
This may or may not be true. It may well be the case that she took the job under a job description that includes providing services to females only. The bigger question is whether she is correct in treating this individual as male and not part of the clientele she signed up to provide services for and or whether this individual should be regarded as female and should be provided services.
I don't see her religion as being an issue to use to oppose her choice.
In most circumstances, I don't believe that people should be allowed to discriminate on the basis of the religion, gender, perceived race, language, etc. of the individual receiving services. However, I do think that there should be exceptions for some intimate services, as well as services requiring specific expertise. For instance, only a fool would go to an ob/gyn for a vasectomy even if the ob/gyn routinely provides sterilization services for female patients. Likewise, an individual who walks into a barber shop looking for a hair color, wave and set for their shoulder length hair is foolish to expect such services and the barber would be foolish to attempt them. Someone who routinely does tattoos or piercings should have the right to refuse to perform a service if s/he feels that the service requested would be unsafe or otherwise result in potential harm to the customer seeking services. Likewise, a tattoo artist should be allowed to refuse to tattoo images or words that s/he finds personally offensive or to refuse to tattoo an individual under the age of consent, even if the parent is there and provides his or her approval. Example: A parent brings their 13 year old in and insists the tattoo artist tattoos the words I Am A Thief across the child's forehead. The artist can and should refuse (and call CPS immediately).
Individuals who provide intimate services, such as waxing, but also a number of other services and even treatments should have the right to restrict their services to clients which are acceptable within the service provider's religious or cultural traditions or sexual preferences. A male prostitute who exclusively services male clients should not be expected to service a female client, for example.
In this particular case, the salon should have immediately provided the client with another person to perform the waxing and then do some investigation into the legalities, the physical issues that might arise, the ethics of providing intimate services to trans individuals and provide further training to staff as to the best possible way to meet the needs of all their clients. I do not believe that any client has the right to demand services from whichever service provider they choose. If I walk into my hairdressers and demand that Cindi* do my hair, I should not expect to have my demand met if Cindi declines to do my hair. And frankly, I would be foolish to make such a demand, if I cared about my hair and how it looks.
*Cindi is fictional.