you are a nonresident alien if you were on J1 for less than 2 calendar years. You would need to file form 1040NR or 1040NR-EZ plus form 8843. These are not supported by Turbotax.

To get a refund of your federal income taxes withheld, assuming you are entitled to one, you have to file a federal income tax return for a non-resident alien, form 1040NR.
To get a refund of state income taxes withheld, assuming you are entitled to one, you have to file a state income tax return for that state.  File as a nonresident.
To get a refund of the FICA and Medicare taxes withheld by anemployer, the first move is to ask for a refund from that employer, on the grounds that the employer should never have withheld those taxes in the first place because you had a J-1.  See IRS Publication 519, section "Social security and medicare taxes", subsection "Students and Exchange visitors."

if the state return is a nonresident return depends on the state. a nonresident alien for federal purposes is not automatically a nonresident for state purposes

True, it's most likely the questioner is a nonresident, but could be a resident, especially if in one state for more than 180 days.