Horrible squeaking is usually from dry derailer pulley bushings, or maybe a chain that's had all the lube washed out of it...

Another possible cause of such squeaking (although thanks to your rigor detailing the symptom, it seems highly likely to be pulley bushings), is a common stumper with Shimano MTB rear hubs, where a rubber boot on the axle sits stationary against the rotating hub; you might have something similar.