Give the Block the "Platform" Behavior. That'll give it automatic gravity too.
Now, for the movement, Use the "Set X Component of Movement" in the Platformer actions to make it move as your character is pushing ("Move Left/Right" Control is Down + Overlapping at Offset (0,1)) the block. Use an "Else" to set the movement back to 0 when you're not pushing it.
Hope this helps.
Here is a example using Xeno's method.
The thing is you can't give something platform behaviour without setting it's controls. So if you give the box platform behaviour and set it to player 2 controls that aren't set in the application properties the box will go mad and fly all over the place. So give it player 1 controls and set it to always ignore input.