Hello,

Private students are the students who do not attend any school. Such students are allowed to appear for the secondary school certificate and higher secondary certificate examinations, provided they apply and submit their Form 17s through schools specified by the board.

The students, who are unable to give the board examination via regular school or college have the facility to appear for both class X and XII board as external students by filling up form number 17.
Students who have completed 14 years of age are eligible for submitting the Form 17 for SSC examination, while those who have passed SSC can submit the form for the HSC examination externally.

So according to these eligibility rules you don't fit these eligibility criterias since you have already given and passed the board exams.