I am not a lawyer, this is not legal advice. Is there any two-year home residency requirement on the J-1 holder's visa (check DS-2019)? That may affect your ability to switch to H-1B (or any other visa).
By default J-2 visa holders have to leave the country?
Short answer - my spouse and I do not fulfill the main conditions they specify for having to leave the country after the visa expires.
Ok, my understanding was that (again, I’m not lawyer, this is not legal advice!), if there is a two-year home residency requirement on the J-1 holder’s visa, then both J-1 and J-2 holders cannot switch to another visa before you have either (a) resided for two years in the country of residence, or (b) were granted a Section 212(e) waiver approved by USCIS. But perhaps that’s different for some J-1 categories.