There is only one grd block in the box. Do I connect the green and White at the same block? Rick

If the panel is the main panel feeding from the meter then the green would go to the ground block and the white would go to the neutral bar with a ground screw or ground strap bonding the neutral bar and ground bar together. If your feeding to a sub panel from the main panel then the green goes to the ground block and the white goes to the neutral bar and NO ground screw or ground strap should be used.

I am going to a sub panel ,but it only has a single block for ground . If I connect the white wire to that, do I just connect the green wire to the side of the box itself or to the block where the white is.

I think what you are calling the ground bar is actually the neutral bar and is not part of the metal panel. Since this is a sub panel the white will go to neutral bar and you will need to install a separate ground bar that attaches directly to the metal panel. Any new romexes you add onto the panel will have the ground wire go to the added ground bar and the white wire goes to the neutral bar.