They're fully within their rights to ask you to remove the nose ring especially if you will be coming into contact with their clients. The fact of the matter is that during business hours you represent them and your image reflects upon theirs. If you wear long sleeves/a suit nobody can tell you have a tattoo, but if you slap a band aid on or makeup on it's still possible for people to see that you have a nose ring due to the fact that no matter how tiny it is it will protrude out. Now depending who the client base is they may be able to notice this and your employer simply does not want that image out there. It's about looking professional while representing the company.
Understand that no matter your views on the matter they still want their image represented in a certain way. As long as that way is not discriminatory they are well within their rights to ask that of you should you take the job. Remember perception is reality as far as most people are concerned, odds are your clients are no different. Due to that, many employers will tend to err on the side of appearing safer especially with how their employees look while representing them. This will range from banks to doctors offices to UPS to food services to IT. Each employer will independently choose as per how far to go with dress code but they'll all have some sort of clauses in the contract about how you appear while you're on the clock. They are perfectly within their rights so long as its not discriminatory and as such there's nothing you can do but choose to either work there and abide by dress code or decide that the job is not for you which you are well within your rights to do. I mean even look at sports teams, some of them have this, an example being the Yankees. Their players sign the contract and its said you need to be clean shaven, always have a short hair cut (no dreads etc), don't show tattoos and these are guys signing up for millions often at the top of their game but the Yankees say you represent us on the field and off so they even add in off field rules to the contract.
Also depending on what it is they expect you to do they can call it a possible hazard to your safety and ask it removed.