Is there any prohibition if an individual opens his store during Eid days?

It is permissible for a Muslim to open their shop during Muslim holidays, provided they do not sell items used for disobedience [to God]. It is also permissible to open it during non-Muslim holidays, provided they do not sell items that assist [non-Muslims] in their acts of disobedience, or items that assist Muslims in imitating them during their holidays; this is based on the verse: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression), and because doing so would be assisting in evil and glorifying their polytheism.

Therefore, it is not permissible to sell products specifically designated for the holidays of disbelievers, such as crosses, statues, and greeting cards. As for other goods that may be used during the holiday and at other times, the merchant must not sell them to anyone whom they know or strongly suspect will use them for forbidden purposes or for celebrating [non-Muslims'] holidays.