iPhone Google Reader client with updating home-screen badge for unread item count?
No Google Reader App that I have used/seen has had an option to do this.  It would require the app to be running in the background and checking, which Apple doesn't technically allow in the case that we are speaking of. This mostly has to do with how Apple gives certain privileges to Apps when multitasking.  It may be something that gets expanded on in the future but I have not seen any apps at the moment that will do what you asking.

I use the Reeder on both my iPad and iPhone and it has an unread counter home-screen badge but it won't update it will my phone is sitting idle.

I know it probably wasn't the answer you were looking for but I hope it helps.  Until Apple allows for better multitasking features to be available to developers, we won't see any apps that do this without extreme work arounds by developers or jail-breaking.  

I don't recommend jail-breaking your iDevice but other will probably mention that it might be possible to do it through that kind of means, but that is your choice to go down that route.