This question appears to be about Customs and Border Protection immigration inspectors rather than Border Patrol officers. (Border Patrol is a distinct agency, although it is subordinate to CBP in the government's organizational chart.)
Are US citizens obligated to disclose foreign citizenships to US border patrol upon entry?
I know that US border patrol has the right to determine an individual’s admissibility to the United States.
US citizens cannot be found inadmissible. Grounds of inadmissibility apply only to aliens. See 8 USC 1182.
given that I am a US citizen, could I be arrested/denied entry if I do not wish to disclose my other nationalities for privacy reasons?
You cannot be denied entry. You can be arrested if the arresting officer has a warrant or probable cause to suspect that you have committed a felony. Now it's possible to imagine circumstances under which your not disclosing additional citizenships might reinforce an officer's suspicion that you have committed a crime, but by itself it cannot provide that suspicion, because of the fifth amendment.
In practice, of course, an immigration officer can make your life difficult by delaying your entry. While they are not supposed to delay you on immigration grounds once they are convinced that you are a US citizen, they can delay you on customs grounds and they could probably even manufacture some uncertainty about your US citizenship if they were pressed to do so. For example, they could claim to be looking into whether you had performed any of the statutory "expatriating acts" that could result in your losing US citizenship (but only if you perform the act with the intention of losing your US citizenship, and the government formally presumes that you have no such intention).
Finally, in short, it's not legal for them to demand an answer to that question, but they can nonetheless make your life difficult if you don't answer it. There's no judge looking over the immigration officer's shoulder, and the only way for you to get the question before a judge is to file suit, a process that takes weeks to years. At the end of a long flight, if you want to get home, your best course of action is probably to answer.