Can a business refuse to take credit cards from some customers, but take them from others?
A credit card is not "legal tender" - - it is merely one of many "convenient" ways of paying a debt. Just because a merchant accepst credit cards for payment of debts doesn't mean that they are compelled by law to do so.
Although there might be other reasons, one reason for refusing credit cards, particularly as payment for services, is when the merchant has gotten the impression that you may not be satisfied with your purchase and might later dispute the charge with your credit card company.
As a merchant, they can demand any form of payment they want. If they demand "cash", that's their right. If they demand payment by way of chickens, or oranges, or gold bullion, that's their right. Obviously, they don't wish to take this chance with you, and are now requiring "secured" payment in the form of a cashier's check.