Yes farm fresh eggs can sometimes give you diarrhea especially if the farm fresh eggs contain some bacteria that the store bought eggs do not.
Sometimes farm fresh eggs may have bacteria that is harmless but can lead to diarrhea and other stomach issues and sometimes you might get diarrhea as a result of the farm fresh eggs having Salmonella.
The Salmonella poisoning can also cause fever, abdominal cramps and yes the diarrhea but if you're just getting diarrhea as a result of the farm fresh eggs then it could be just the farm fresh eggs causing the diarrhea.
Some people only get diarrhea when eating farm fresh eggs and can eat the store bought eggs without diarrhea happening to them.
But if you continue to experience diarrhea as well as the abdominal pain etc then see a doctor to make sure it's not Salmonella.