Do there exist non-religious or non-christian writers or philosophers that analyse Atheism from a skeptical point of view?
Is it even possible to skeptically analyze something that (at it's core) is a rejection of a non-evidence based assertion?
Typically, the thing to be skeptical of is the positive assertion (i.e. "there is a god"). The rejection of an assertion due to a (perhaps perceived) lack of evidence is rationally the default position. You don't start off believing everything you're told until someone disproves it, otherwise you totally NEED to paypal me $100 right NOW or the world is going to end. I'm not even kidding! Quick, or it's all your fault!
Really, though, you can find some skeptical discussions on "hard atheism" (e.g. "there is no god"), but that's pretty light work. Being an atheist myself, I have trouble even imagining what absolute proof that there is no god would consist of. There are so many definitions of "god" that ruling them all out is effectively impossible.
Yes. A Hindu philosopher by the name of Gaudapada. He wrote a commentary on the Mandukya Upanishad called Gaudapada's Karika. It is a defense of the Hindu Advaita (non-dual) philosophy. Chapters 2, 3, and 4 specifically are arguments defending the non-dual Brahman using reasoning and logical argumentation alone, no scriptural references. In those chapters he gives opposing arguments of nihilists and materialists and defends his arguments against those opposing views.
Probably the classic skeptical treatment of atheism is T.H. Huxley who coined the word 'agnostic'.
You will also find a fair number of scholarly articles critiquing certain kinds of atheism if you search for pairs of terms like "soft atheism" and "hard atheism" or "weak atheism and "strong atheism" or "negative atheism" and "positive atheism". The weak/soft/negative position is that person x lacks a positive belief in God while the strong/hard/positive position is that person x believes that God does not exist. The latter is a far more sweeping claim. Michael Martin has a number of writings discussing the difference.