What would cause an ADC to stick on an arbitrary value?
This ended up being a software problem. The HX710 has two modes: A) differential input B) DVDD-AVDD measurement. You set the mode for the next conversion period by the amount of PD_SCK trailing pulses in the current conversion period. This is shown on page 5 of the datasheet:  



In the driver code, I neglected to disable interrupts during the data clock out period. This was the cause of the intermittent failures. It seems that interrupting during the clock out sequence would sometimes cause the ADC to begin reading out DVDD-AVDD instead of the differential input. That explains why I would see it hang at an arbitrary value not responding to any inputs from the load cell. 

Disable global interrupts while driving time sensitive peripherals!

Lesson learned.