I
compost all eggs in that condition. Better safe than sorry, in my opinion. When I find eggs in dry places that seem like they have just been laid within the last week, I put them in a bowl of cool water. If they float I throw them all away. If they all sink I generally use them to feed piglets or chickens. As an egg goes bad it produces gases inside that cause it to float.
Wet and dirty... cold or not, compost pile. I don't even break them. let that happen in the pile. Bad smells will abound from one rotten egg. YUCK!