The first day of the Christian year should be regarded as an ordinary day. We should not show any sign of celebration. Everyone should check his family and children lest they celebrate it due to the influence of a friend or a TV channel, or due to a sense of curiosity or imitation. It was narrated on the authority of Ibn ‘Umar, may Allaah be pleased with them both, that the Prophet, sallallaahu ‘alayhi wa sallam, said: “Each of you is a caretaker and each of you is responsible for those under his care. The ruler is a caretaker of people and is responsible for his subjects. The man is a caretaker of his family and is responsible for his family members. The woman is a caretaker of her husband’s house and children and is responsible for them. The servant is a caretaker of his master’s property and is responsible for it. Each of you is a caretaker and is responsible for those under his care.” [Al-Bukhaari and Muslim]
Ibn al-Qayyim (may Allaah have mercy on him) said: it is not permissible for the Muslims to attend the festivals of the mushrikeen, according to the consensus of the scholars whose words carry weight. The fuqaha’ who follow the four schools of thought have stated this clearly in their books… Al-Bayhaqi narrated with a saheeh isnaad from ‘Umar ibn al-Khattaab that he said: “Do not enter upon the mushrikeen in their churches on the day of their festival, for divine wrath is descending upon them.” And ‘Umar also said: “Avoid the enemies of Allaah on their festivals.” Al-Bayhaqi narrated with a jayyid isnaad from ‘Abd-Allaah ibn ‘Amr that he said: “Whoever settles in the land of the non-Arabs and celebrates their new year and festival and imitates them until he dies in that state, will be gathered with them on the Day of Resurrection.” (Ahkaam Ahl al-Dhimmah, 1/723-724).
In the light of the above mentioned reasons, Muslims can't eat nuts, turkey etc in New Year's Eve with the aim of celebration of this day. Otherwise, they can eat whatever they want if they don't think to celebrate New Year's Eve. Also, they should avoid doing something that Christians do in New Year's Eve.