I recently created a character where the hair, shirt buttons, nametag (he's wearing a uniform), &c are separate objects. I parented (as opposed to grouped) all of the head objects together while modelling. I haven't yet bound the head, so I can't answer that, but I bound everything else to the skeleton successfully (more in a minute here).
I think the idea is NOT to select the group but to select every single object WITHIN that group and then bind. Be sure and freeze transformations on anything being bound!
Well, as I said I got all of the body objects bound to my skeleton. When it came to moving the joints around prior to painting weights, I noticed that while objects (like buttons, nametags, &c) move with the rest of the figure, they don't always move as desired. Take the buttons, for example.
When I set up my smooth bind parameters, I set the max influence to 2 and the dropoff rate to 8 (as per Mike McKinley's EXCELLENT character rigging tutorial, which I HIGHLY recommend purchasing off this website if you don't already have it). This means any given object will be influenced by up to two adjacent joints on either side of it. Since buttons are small and don't necessarily line up perfectly with joints on a skeleton, they will deform when I try and bend the skeleton. To fix this, I had to "paint the weights" so that the button in question was influenced by only ONE joint. This keeps the button in one piece when it does move but also keeps the button from moving naturally.
What I am going to try (and I will report back) is to fix this by creating blend shapes. Since the buttons are separate objects, I imagine they can be grabbed (in component mode) and moved to wherever they need to be moved to (again, Mike's rigging tutorial will make all of this MUCH easier to understand). If this works, then it should work for your character as well.
If for some reason it doesn't work, constraining the object to a skeleton joint might.
The important thing to remember is that you want to bind each object in the group to the skeleton. You don't have to bind everything at once, either! BTW eyes are usually manipulated by constraints, usually Aim constraints.
While this may not completely answer your question, I hope it helps and I will report back with the result of my own efforts and share it on this thread...
"Ad astra per aspera..."