Undercooked eggs that were in the pot plenty long I'm seeking information because oddly this has recently also happened to several of my friends. 

I cooked my hard-boiled eggs like I normally would; they were in the pot until the water boiled and I even left the burner on low while I ate dinner. I tend to over-cook them anyway, so they were in there at least 20 minutes. I pulled one out of the refrigerator this morning and it was very soft-boiled; even the white was gooey. What could be causing this? These eggs are not old, but could they be spoiled?
If your eggs came out soft boiled then that means they weren't exposed to enough heat - something in your method went wrong. I suspect that when you turned it down low the water temperature dropped below 140-150 degrees F, which is the temperature that egg white will cook. It doesn't matter how long you cook eggs below 140F, they will never get done below that temperature. 

To fully cook egg white it must be heated to 150F, egg yolk to 160F. If your eggs were cold when you put them in and the water boiled quickly the inside of the egg would still be fridge temperature, so when you turned the heat down low the eggs cooled the water below 140F, and the amount of heat the burner was producing wasn't enough to heat the water and eggs above 140-150F.

I use a similar method, however I put a lid on the pot which makes a big difference. If you don't want or have the right lid simply keep the heat turned up to where the eggs bounce around a bit.