What is the ruling on my sister's unregistered marriage (zawaj urfi) and her pregnancy from this marriage? Is it considered a legitimate marriage, given that the husband does not want to acknowledge the children? And is it forbidden for me to boycott her because of this?

If the marriage was conducted with the permission of the guardian (wali) and the presence of witnesses, then it is valid. If it was conducted without a guardian, in accordance with the Madhhab of Abu Hanifa, or if a judge who deems it valid ruled it as such, then it is also valid. It does not matter if it is a customary marriage (ʿurfī), meaning it was not officially documented. However, if 'customary' means merely an agreement between the two parties, then it is not a valid Shar'i marriage. As for the children, if the marriage is valid, then they are his children. If the marriage is not valid, they are still attributed to him if he believed the marriage to be valid. The husband's denial of his children's lineage is a grave transgression. As for not officially documenting the marriage, there is no issue with that. Boycotting the sister is impermissible if her marriage is valid or if she had a valid excuse. It is permissible if she proceeded with the marriage knowing it was invalid. Prioritizing benefit is preferable.