I know that the terminology of “public” school is differently between the U.K. and the U.S. - I don’t know if the same applies to India.
So to clarify - in the U.S. the “public” schools are the local schools run by your city of residence. The funding of the school (and with that, how well they are equipped, what staff they can attract/retain, what advanced course levels they can offer,…) comes from the town’s property taxes, thus is closely related to the value of private residences, and if they can attract high-value businesses. Consequently the “better off” an area is, the more likely it can afford to fund “good” schools.
In NJ you don’t get to “choose” a public school, making your question moot. Your public high school will be assigned based on the address in town where you live and pay your local taxes - although some larger cities do have specialized schools to serve gifted/talented students.
There are a very few cases of little towns not having their own high schools, but instead they are sharing costs with neighboring towns. In rare cases, students may pick from two regional high schools if their town is paying to two different towns.
Bergen Academies is a “public” high-school run be Bergen County for it various towns. Yes, it is highly regarded, highly-selective - and not open to out-of-county residents.
So - what you are likely looking for are “private” schools. While the majority of them will have a religious affiliation, don’t let that scare you off. They don’t discriminate based on faith, as they are very much depending on parents of any faith willing to pay substantial private tuition year after year. A good number of private high schools will be run by the Catholic Diocese, some of them will be co-ed, others will be male or female only.
Finally, there are also boarding schools, many out-of-state.
Now - whether any of that will buy you a ticket to MIT or Princeton is highly questionable. Colleges will not be majorly impressed by what high school you happened to attend for grade 11 and 12. More important will be what courses you took, if those courses were “honors” or “AP” level courses, what grades you earned etc.
The quality of school will be important, because it might control how broad their course catalogue is, how many honors and AP classes they offer, etc.