Are there any Sharia texts that support the position of a girl who refuses to shake hands with her brother-in-law (who is also her cousin) when her family pressures her to do so and accuses her of being rigid?

The questioner did not specify the degree of kinship of the son-in-law. If the son-in-law is not a mahram by blood,

or by suckling, or by marriage, then it is not permissible to shake hands with him, or to be alone with him,

or to appear before him without proper Islamic attire. The sister must remain steadfast in righteousness and obedience

and not succumb to pressure. She must be patient with what she faces, for whoever pleases Allah,

even if it displeases people, Allah will suffice him. Guardians should rejoice in their daughter's righteousness

and support her, not force her to shake hands with non-mahram men.

The sister is advised to invite her family to good with wisdom and beautiful admonition.