So, it is quite clear that Gurren Lagann was a response to the questions raised by Neon Genesis Evangelion. However, before I state the obvious about how Gurren Lagann was the anti-evangelion I'd like to (a) Explore the ways in which the shows are similar and (b) Hopefully get you interested enough in the philosophical topics that you decide to pick up the books dealing with the topics.
[Relatively important spoilers for Gurren Lagann and Evangelion]
Firstly, the similarities.
Both the series are anti-authoritarian in so far as they question why Authority exists in the first place as well as why we should follow authorities in the first place from the first episode. Shinji questions why he should do what his father says him to do and Kamina questions the village chief in a similar manner. But at the same time, neither show is anarchist in the sense that it believes all authority is bad. Gurren Lagann in particular has the character of Kamina which gives us a model of what authority should be. But what is more interesting is Eva, and I will get to it in the next para. Another similarity between the two is that both are what many call "Existentialist". I.e. both shows support the idea that Human beings are ultimately defined by themselves. They give their lives own meanings and that race, gender, etc only defines us if we let them.
Now, why is this important?
Because this helps us better understand how they are different and why Gurren Lagann is anti-Evangelion. Evangelion believes that life is given meaning not by itself but rather how it defineself "itself" to the "others" around them. In ep 25 of Evangelion, when Shinji enters the Human Instrumentality project he is transported to a world without anything, where he is simply floating amongst nothingness a place where Shinji is, according to the show, "Absolutely free". But Shinji doesnt like absolute freedom, it makes him uneasy so he is given a ground to stand on. He loses some of that freedom, but he loses some of his anxiety, he may not be able to float anymore but he can walk, he can interact with the ground and that puts him at ease. For eva, the meaning of life is like that ground. In isolation, a human being is absolutely free but human life is utterly meaningless but human beings can by interacting with other human beings and forming relationships with them, give their lives some semblance of meaning but you can only do so by giving a small part of yourself to another person, letting them define your life in some manner thereby giving up some of your freedom. But Eva sees this tradeoff as over worth it and characterizes being lonely as almost equal to being a nihilist i.e. self-absorbed, endless consumption, aimlessly living etc.
Gurren Lagann on the other hand, believes that humans are absolutely free and there's literally nothing that can take that freedom away. That life is, if we are being perfectly honest, pretty meaningless but WE OURSELVES (not others, Ourselves) can give meaning to it. And that, even if life is meaningless, we must make the impossible possible, and carve our own meaning into it. And we can start by rebelling against those things which try to force their own false meaning onto us. The things that tell us "You should do THIS.", "You should become THAT", etc.
Conclusion and TLDR
Eva believes that freedom and meaning are not compatible with each other and that to get the meaning of life we must relinquish our freedoms to a certain degree because we can only attain meaning by defining who we are in relation to other people. Thereby giving others a little bit of power - and dare I say Authority - over us. A philosophy that is similar to Jean Paul Sartre's "Humanist Existentialism".
Gurren Lagann believes that freedom and meaning are not only compatible but are dependent on each other. Meaning is not sth that you are born with, but rather that you give to yourself which is sth you can do best when you dont have someone ordering you around and telling you what you are and should do from above. And as such, you should rebel against anything that tries to oppress your ability to define yourself. A philosophy which is right out of the book of Albert Camus's "Absurdism" (A name which fits Gurren Lagann perfectly.)
Interestingly, Sartre and Camus were philosophical rivals and often shit-talked one another "politely". I hope I got ya'll interested in their ideas and if you wish to, please check out the works that deal with this topic in particular: The Myth of Sisyphys by Albert Camus (Absurdism) and Existentialism is a Humanism by Jean Paul Sartre (Humanism)