Why won't my boiled egg whites harden?
This is not a real answer, but rather some info to hopefully, help:


Egg whites are 90% water, when they harden, the protein unravels and creates a grid network that does not allow the molecules to slip past each other.  If the protein level drops, it might make it harder to set and stay runny.
Egg white proteins go through a phase change (become hard) at a lower temperature than the Yolk by a few degrees, that's how you can have soft-boiled eggs.
If you crack a fresh egg and it sprawls on the dish (as opposed to hold tall), you're looking at an egg from a chicken that didn't eat many insects/larvae. ;) This may be expected as at this time of the year (northern hemisphere) insects aren't very active.


There is a good section in Michael Pollan's Omnivore's Dilemma that gets into more detail.  And perhaps better information regarding the off-season feed.