Well the big one by Butler is "Gender Trouble: Feminism and the Subversion of Identity" but I like her recent book "Precarious Life" because I'm a big anti-anthropocentrism fan and she takes up the question of nonhuman animals. Buuut it's kinda helpful to have some background in poststructuralism. You could look into Derrida and Foucault. Where the hell to start with Derrida tho? I just kinda jumped headfirst into "Of Grammatology".
Arendt is really well-known known for her work on fascism and totalitarian regimes, especially since she experienced Nazism first hand. If you are interested in this "The Origins of Totalitarianism" is the book for you. I'd recommed reading Walter Benjamin's "The Work of Art in the Age of Mechanical Reproduction" since Benjamin was a major influence on Arendt and you can see this in her writings. Giorgio Agamben's "The Open: Man and Animal" is also useful after you've read the other two. It is helpful to keep in mind that these writers are all very stylistic and bring concerns of aesthetics to bear in political life, so you shouldn't go into it expecting a very clean cut, dry academic essay.

There's some very contentious claims that Arendt makes in "On Human Nature" on the topic of labor and work and "the public and private realms". More or less she can be read as defending the preservation of a structure of public/private life that mainly originates out of power relations between man and woman and is critical of governmentally instantiated social programs given they, according to her, undermine the distinction and thereby threaten freedoms built upon it. (Presuming of course that there really is a distinction between labor and work, which sounds awefully essentialistic and probably not at all reflective of the complexities of both, which is where Butler would step in and say naaaaahuhh these contexts are always different and you can universalize that distinction.) When read alongside the consideration that some of these programs help bring about women's equality in the workforce, things get a bit... icky.