State income tax refund?

You have to file a state income tax return along with your federal income tax return. Both are completely separate, and a refund on one never implies a refund on the other. You state will have a different tax rate than the IRS has, and different tax brackets.

I have never used JH, whether online, in their office, or at Walmart, but most other online sites I've seen allows you to file a state return also, so I can only assume they do, too. How you do it should be like any other site - Just follow the instructions and answer the questions.

Depends on your state whether you can get it refunded.

depends on which state and what the standard deduction is for that state and your status
if you filed with J/H, you would need to file the state return with them as well