Any device used again and again, egg after egg, is a cross-contamination risk, spreading the "ick" on one, potentially, to many others.
The same could be said of the scrubby, washcloth, loofa, whatever we use to clean our eggs. Or the water used to wash or rinse them.
One of the reasons the USDA wants us to spritz with sanitizer and air dry - not dunk in sanitizer - after cleaning the egg (and removing much of the bloom).
Is it possible??? YES.
Is it likely? Depends on your other sanitation.
Like everything else, its a matter of risk management. I'd favor an alcohol based, food grade ink *IF* I was to do it.
That said, WHY do it? Branding and Date coding are things I can do with my plastic egg flats - which are both reuseable and easily sanitized - which is why I bought plastic egg flats. If you are in a "carton" state, the same can be done on your cartons.
So I would not do it, because it serves no purpose that I am aware of.