Is it permissible for a woman to take off her Hijab (veil) in front of the maternal or paternal uncle of her mother or father (i.e. Her maternal or paternal great-uncle) and shake hands with them?
What is the Fiqhy (Islamic juristic) evidence for considering this Halal (lawful) or Haram (prohibited)?
It is permissible for women to show in front of the maternal and paternal uncle of their mothers or fathers the same adornment that they are allowed to show in front of their Mahrams (unmarriageable relatives) for two reasons:
They are Mahrams to them, for it is prohibited on each of the two great-uncles to marry them due to the general prohibition in Allah’s Words:
Forbidden to you (for marriage) are: your mothers, your daughters, your sisters, your father’s sisters, your mother’s sisters, your brother’s daughters, your sister’s daughters
(Part No. 17; Page No. 386) The term "your brothers’ daughters" in the Ayah (Qur’anic verse) does not just refer to full brothers’ daughters, but embraces brothers’ daughters and their descendents. The paternal uncle of a woman’s father is her grandfather’s brother, and grandparents are ruled as fathers no matter how high the pedigree ascends. Therefore, this woman falls under the general prohibition on marrying brothers’ daughters. Also, the paternal uncle of a woman’s mother is the brother of her mother’s father; therefore, she is included among his sister’s daughters. On the other hand, "your sisters’ daughters" who are referred to in the Ayah includes not only full sisters’ daughters, but embraces sisters’ daughters and their descendents. The maternal uncle of a woman’s mother is the brother of her maternal grandmother and the maternal uncle of a woman’s father is the brother of her paternal grandmother; therefore, this woman falls under the general prohibition on marrying sisters’ daughters. Accordingly, if a woman is proven to be one of the Mahrams of the great-uncles mentioned in the question, it becomes permissible for her to show in front of them what she is permitted to show in front of the men mentioned in Allah’s Saying:
...and not to reveal their adornment except to their husbands, or their fathers, or their husband’s fathers
Allah (Exalted be He) has permitted women to show of their adornment in the presence of their brothers’ sons, their sisters’ sons, and their descendants as much as they are permitted to show in the presence of their other Mahrams, such as their fathers, sons, and brothers. Allah (Exalted be He) says:
...and not to reveal their adornment except to their husbands, or their fathers, or their husband’s fathers, or their sons, or their husband’s sons, or their brothers or their brother’s sons
(Part No. 17; Page No. 387) No matter how high the ascendant generations extend, the maternal and paternal uncles of a woman’s parents are included in the lineage relationship towards the children of one’s brother or sister, no matter how many descendant generations branch down; therefore, the ruling on showing adornment in front of any of them is the same; as for their greeting them, it should be by shaking hands only. May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.