Is it permissible to shake hands with a paternal cousin (a male's daughter of his paternal uncle) in front of family and without ill intention, especially when she extends her hand to greet, during a visit to a sick uncle?

Shaking hands with a non-mahram woman is absolutely impermissible, whether her mahrams are present or not. The prohibition becomes more severe in their absence. You must inform your cousin and her family of the impermissibility of this act, for shyness in this context is blameworthy, and obedience to Allah comes first.