Paula Kirby has a nice article on the Washington Post site about Christians who claim to be persecuted just because they are required to obey the general law that applies to everyone else, or because they are required to conform to work requirements under their contracts of employment - the same work requirements as apply to everyone else.
Historically, this was not what persecution was about. In principle, if you are simply being required to meet some requirement that is imposed for a good secular reason that would exist whether your religion did or not ... you can't claim that this is persecution. The intent of the requirement is not persecutorial, but is, rather, blind to your religion. It is not the same as someone setting out to suppress your religion or to impose on you a religion that you don't accept.
It may, of course, still be important to you. There may be reasons why the people imposing the requirement should compromise if it causes you all that much pain and anxiety, and if the compromise will not do too much to undermine the purpose for which the requirement was implemented in the first place. Anti-discrimination law may categorise the requirement as indirectly discriminatory if it is imposed by an employer - though merely indirect discrimination can usually be justified if there is a reason for it genuinely based on business efficacy. (You can guess, of course, that this issue gets a fuller discussion in Freedom of Religion and the Secular State - my background in employment and labour relations law helped here).
In any event, even if some compromise is offered ... compromise is a two-way street. You don't get to complain about being persecuted (well, you can because you have freedom of speech, but no one need take you seriously) just because you're required to abide by the same rules as everyone else. And you certainly don't get to complain if a compromise is offered to you but you are the one who is not prepared to compromise.