Hello,
NEET entrance exam is taken by students throughout india for admissions to medical/bds and allied health courses.
Yes you are eligible for admissions to private medical institutions outside J&K after applying for 100%state quota.
You are just not eligible for admissions under 15% all india quota seats.

Yoy can find the colleges you can get at your NEET score using our college predictor :- https://medicine.careers360.com/neet-college-predictor

Hope this helps