A lot of my friends studied NERIST. According to them, it offers admission in three modules, Certificate, Diploma, and Degree programs, based on the entrance scores in NEE 1, NEE 2, and NEE 3, respectively.
The eligibility for taking the NEE 1 exam is to pass matriculation or 10th class. It offers admission in the certificate program equivalent to ITI or 10+2 class as declared by UGC.
Therefore, if you take admission through NEE 1 in NERIST, two years down the line of studying the basic module, you will be equivalent to an HSC student. So, yes, you can appear for JEE.