My question is a kind of follow-up to this. When apples have turned soft or have started to shrivel, is it possible / advisable to but them through a juicer? From the linked question is would appear that they have started to dehydrate, so I wouldn't expect to get much juice from them.
It's not going to hurt anything. The apples they use for cider are usually pretty rough, so a little wrinkling isn't going cause a health issue. You very well may not get as much juice, but the juice you will get will be more concentrated.
The same principle applies to grapes used for wine...Ideally they will get very little water in the weeks leading up to harvest, so you'll get all the good sugars and flavours, and less of the undesirable water.
It depends on the type of juicer you use. If you use a centrifugal juicer, or a slow juicer, you will find you get less juice. I'm not sure why, but softer apples result in less juice from these types of juicer. If you use a cider press (and muslin cloth bags), you will find you get results that are about on par with fresh apples.
It's worth noting that, in old, bruised apples, mould tends to grow quite quickly inside the bruised flesh. If there is some mould in the early stages of growing (i.e. too small for you to see), you will then introduce that into your juice, and into your brew. This will taint the whole batch, resulting in cider vinegar, or a big mass of mould.