`Eid Al-Fitr is just one day on which it is prohibited to fast. As for the second and third days of Shawwal, it is not prohibited to fast on those days. It is permissible to fast it to make up a missed Ramadan fast or as a voluntary fast.
The date of ‘Eid al-Fitr is determined by sighting the new moon after the month of Ramadan. So there’s no issue if ‘Eid al-Adha falls on Friday in one country and Saturday in another, and so on—because these differences happen due to variations in moon sighting.
The Prophet (peace and blessings of Allah be upon him) encouraged seeking the sighting of the moon .
How ‘Eid is determined (moon sighting):
Islam follows a lunar calendar, so months begin with the sighting of the new crescent moon (hilāl).
‘Eid al-Fitr happens when the moon for Shawwāl is sighted (the month after Ramadan).
Because the moon may be seen earlier or later in different places (weather, horizon, location), countries can legitimately have ‘Eid on different days.
If the moon is not seen, the Islamic month is completed as 30 days, then the next month begins.