Do I need to be on H1B to apply for green card through employement?

Your employer can begin the process now.  The problem with being in J2 status is that visa type doesn't allow you maintain non-immigrant status while applying for a green card.  As long as you remain in the US, it won't be an issue, but if you leave and try to re-enter in J2 status after your employer has filed an immigrant petition for you, they will likely turn you away for immigrant intent and you would have to process your immigrant visa (if approved) outside the US.  The H1B visa is a dual-intent visa and wouldn't present the same problem if you left the US after the I-140 petition had been filed and tried to return.  In H1B status, you may simultaneously seek immigrant status which isn't true of the J2 visa.