In Chicken Run we see that Rocky runs after Ginger into the pie machine and manages to hold on to some overhanging rod to escape the mincer. Does Ginger also escape the same way? Does that mean the mincer is ineffective and any chickens can escape being minced?
Right after Rocky misses Ginger, he steps on the "Veg(etable) Feed" lever.
This causes a bunch of...
...to fall on his head and drag him into the machine.
As he slides down the chute, he comes to a junction that says "VEG" to the right. This then flips over to say "MEAT" on the left, and he goes down that way. So it appears that he inadvertently diverted Ginger away from the mincer by making her go into the vegetable chute.
Rocky luckily managed to grab a rod that for some mysterious reason is in the machine. The presence of that rod is most likely an error on Mr. Tweedy's part to have left it in there, because if it falls into the mincer it should seriously damage the blades. Lucky (again) for Rocky, it does not do that but gets stuck halfway down and lets him slide to (relative) safety.
Would this allow all chickens to escape? Well... no, not really. As the clip shows, Rocky manages to grab and hold on to the with one wing out of pure desperation. This is quite an athletic feat. Note though that Rocky is much leaner than the chickens, and can easier carry his weight. With the exception of Bunty (shown earlier to do one-winged push-ups) we can expect most chickens to not be able to pull that same feat off. Instead the best they can hope for is bouncing on that rod, and then they are likely to plunge right into the blades.
The only thing this leaves unanswered is how Ginger escaped being diced like the vegetables that later gets dumped on them while in the pic crust. But since chickens were not supposed to go down that chute anyway we can simply assume that some feature of the vegetable chute simply allowed her to get away from that part.