Is it forbidden to boycott neighbors and not visit them if visiting them leads to committing many sins like backbiting, and if they criticize and mock, which compels me to backbite and dislike them, all in avoidance of committing sins?

The rights of neighbors are great, encouraged by Islam, and enjoined by the Book of Allah and the Sunnah of His Prophet. Allah Almighty says: (Worship Allah and associate nothing with Him, and be good to parents, and to relatives, and orphans, and the needy, and the neighbor who is near of kin, and the neighbor who is a stranger, and the companion by your side, and the traveler, and those whom your right hands possess. Indeed, Allah does not like those who are conceited and boastful). And the Prophet, peace and blessings be upon him, said: "Whoever believes in Allah and the Last Day should not harm his neighbor." And he said: "None of you truly believes until he loves for his brother what he loves for himself." And he said: "Jibreel kept enjoining me concerning the neighbor until I thought he would make him an heir."

The mockery of neighbors is a sin from which they must repent. Allah Almighty says: (O you who have believed, let not a people ridicule [another] people; perhaps they may be better than them; nor let women ridicule [other] women; perhaps they may be better than them). And the Prophet, peace and blessings be upon him, said: "O Muslim women, let no neighbor despise her neighbor, even if it be a sheep's trotter."

One must be kind to neighbors and guide them to goodness with kind words and purposeful means. If that is not found, and sitting with them leads to sins like backbiting, then there is no blame in shunning them and turning away from them.