Is leaving eggs out of the fridge, prior to cooking them a direct cause as to how shell break?

Assuming your fridge temperature is around 4 C (39 F) and your room temperature is 22 C (72 F) then that's a difference of 18 C or 33 F. The two temperatures are not particularly polar so why should it crack differently to when it comes straight from the fridge? It won't be like if you threw hot water onto a frozen window where it would crack, the 2 temperatures here are a lot more polar, 1 below freezing point and 1 near boiling point.

The change in temperature would be fairly gradually (I don't have any Scientific evidence for this but I know from experience as when I take an egg out of the freezer and wait for it to reach room temperature -for culinary purposes- it takes around 45 minutes).

Also on a chemical level, an egg shell being mainly calcium carbonate there is no reason for such a gradual change in temperature to weaken its structure especially as it is in the form of a very stable and strong shape. So no is the answer, I don't think it makes any difference in the weakness of an egg or how it cracks!

Hope this helps!

Aside from technique (always on a flat surface, never on a knife or bowl edge) the age and temperature affect the cracking due to moisture and elasticity.

As eggs age the membrane separates from the shell which leads to smaller bits being able to fall away.

An egg which is fully at room temp, which is more than a few minutes, will have more elasticity to its membrane. This will worsen shell chip loss for older eggs. For decently fresh eggs it will not matter.

Bottom line if your eggs are fresh and you crack correctly temp does not matter.

Incidentally if your eggs have a flat spot when hard boiled they are quite old or have been stored improperly.

Maybe.

The reverse seems to be true (room temp eggs not cracking well after cooling), so temp does something - I had some eggs sitting at room temp for a couple days (trying a different preserving method), and used a couple at room temp - not much different from normal eggs, maybe a bit less brittle. A couple others that didn't get used went back in the fridge afterwards. When I went to crack them today - they would not crack cleanly at all, the membrane had to be torn by hand to separate the cracked shell and it still came away in chunks rather than clean halves like I usually get (leaving aside "correct" cracking method, mine is consistent and got different results this time). The white was also much more gelatinous, and stuck more firmly to the inside of the shell and had to be scraped off.

Possibly the difference might come from the change in temp, several times. Maybe the speed of cooling was to blame for the difference (I imagine eggs cool quicker in the fridge than they warm at room temp). If so, a fridge-temp egg that was warmed too quickly might show the same problems. Or an egg that had been stored inconsistently in the past, allowed to cool and warm several times. Or, less plausibly the egg might, ah, adjust to the new temp after a while (so a few hours to days after a temp change are tricky, several days after either way is fine). That could account for why someone might find warming an egg to room temp, harder to crack.

For myself, I can only say the warmer eggs didn't give me any problem - they were a bit less brittle, which meant a tapping out a cleaner break rather than the shell shattering or chipping under the force, and a bit easier to crack (less force to get the cracking started).