Is there an issue of publicly protesting a Jewish business, such as by picketing or boycotting, being a chilul Hashem? Are there any other (halachic or other) problems? What if you believe that the actions of the Jewish business constitute a chilul Hashem?
(This answer is to a previous version of the question, though it has some relevance to the current version also.)
The Bavli (Arachin 16:2) says:
Whence [the rule] that if one sees a despicable thing in his friend then he must admonish him? ― It says [in Vayikra 19:17], "Admonish".… Is it possibly true even if his face changes [from being shamed in public by the admonishment —Rashi]? ― It teaches us, saying [ibid.], "do not bear a sin because of him".
This is codified in Rambam also, who adds:
What are these things said about? Matters among people. But about divine matters, if he did not change his ways privately, we embarrass him publicly, publicizing his sin….
I see no reason to think a business owned by a Jew is different: after all, when you protest the business's action, what you're actually protesting is its proprietor's.
Of course, for practical guidance, consult your rabbi rather than relying on what you read on this site.