Egg shells speckled after cooking
As you posted a high-res picture, I can clearly see that this is exactly like my grandmother's white eggs would look like after she boiled them in an aluminium kettle.  AFAIK they don't sell these any more, so I'm surprised to see eggs like this again after 30 years.

After peeling, the discolouration wasn't noticeable any more.

If both assumptions above (Aluminium pot and not visible any more after peeling) they are safe to eat as Al2O3 is probably the root cause and it's safe according to this document published by the WHO.