I've tried with an external reference voltage myself (from an LM4040, for testing), and I still observed the 50 Hz noise.
For comparison, I also connected an ADS1115 via I²C, and - even with everything operating on battery! - was able to spot the 50 Hz noise there.
So, yeah, the stuff simply is so sensitive that the 50 Hz noise from the environment is being picked up. I doubt the Due construction is to be blamed for that.
Frankly, I was amazed, I (as a software guy) always thought "electro smog" was somet new-agey hippie crap, until it showed up in my own measurements ;-)