Can eggs boil indefinitely?
Boiling means dynamic evaporation of liquid. 

Infinite boiling would mean to have infinite amount of that liquid and therefore the egg would have infinite mass. 

Such an egg would collapse into a black hole and we would not be able to continue in our observation.

In reality, in inert atmosphere, with overlapping stages:


protein coagulation
water boiling
organic matter thermal dehydration
general organic matter decomposion.
carbon stage


If we kept temperature just above $\pu{100 ^\circ C}$ to keep boiling,  stage 3-4 would be very slow and we would not reach carbon.