I think lying down HELPS the stones come out but it is not do-or-die neccesary. I think the lying down is so as not to impede the exit of stones as they are marching out of the gallbladder and liver. Other movements may halt this flow or interrupt it.
For my recent flush, I suffered from insomnia and I am such a bad directions follower I was bopping around the house all night, up, down, up, down. And I got a ton of stones out -- even way more than I expected.