Slider switch needs improvement...
I've had my Mophie Juice Pack Air ever since it got released. I've had no problems with it nor did the mute switch fall off like that.
Only way in can see the mute switch falling apart like pictured above is due to abuse (drop, bang something solid, or pressure on the button causing it to break).
Maybe you slid the phone in the wrong way or there was something on the mute switch when you installed the phone causing it to break?
I have the Juice Pack Plus, which I just got for the 5S. I'm not sure about Mophie introducing a new product just for the 5S, but this works fine. I've had no problems with wireless syncing, the headphone adapter works fine (sound doesn't cut out when jiggled, but maybe it will after a lot of use), and variety of micro USB cables work well with the case.
The slider switch could be improved. Better yet, removed (even if you needed a huge cut-out in the case). All things considered, given the size of the switch and force needed to more it, the slider switch on the case works fairly well, but I could see it breaking after a while after a lot of use, even if not abused. Hopefully, if that happens, it will be within the one-year warranty and Mophie will replace it.
For the OP, I'd try to get a warranty replacement.
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Yes they work with glass screen protectors
They clearly do (I use a non-glass Spigen protector, however), and I'm not sure why Mophie says on its website that their cases are not compatible with screen protectors. On the Juice Pack Plus for the 5/5S, the tolerances are pretty tight, and installed property, the case should not interfere with a screen protector.
One thing, however, is that the lip of the case is not that high. I suspect that it would still be higher than a glass screen protector (particularly one of the new thinner ones that Spigen sells), but it won't provide as much protection as some other cases might. Still, may give it a shot because I really loved the glass protector I had on my 4S.
