Do it right the first time. Do not join the white and bare ground anywhere except at your incoming service at the meter. This is a common cause of stray voltage.  For 220v, run a black, red, white and ground to the shed. White and ground go to separate bus bars in both boxes. The white bus bar should be isolated from the box. The ground bus bar is tied to the box.  Worth hiring an electrician to at least check your work before activating. You don't want stray voltage problems.

I would run wire larger than required for 60 amp to the shed to allow for future increase.