Foreign Income on J-Find

I know the J-Find (jobhunting visa) allows you to do work and includes foreign companies I think. I was wondering if in the first year when you are considered a non tax resident, if you would have to pay Japan taxes on that income. All income would stay in foreign bank accounts and would be utilized through credit cards.

To follow up on that, if no taxes need to be paid in Japan then would there be any potential legal problems if I didn't tell my employer I left the country (assuming no government work/export controlled information).