The first point of the issue is the concept itself of God. The focus of this will be on an existing and willing (having volition to will things into existence and can manipulate things by will) God - the theistic God. A deistic or pantheistic God each refers to a God that works with or is Nature - an anthropomorphic God which is a poetic expression as Nature is the main factor in life. The theistic God is not only apart from Nature, but can violate Laws of Nature at will: e.g. turn water to wine, cause the moon to stop in the sky, move mountains, etc - all by mere will. If one takes the deistic or pantheistic God as the same, granting their respective God the ability to have volition and will, then they give themselves the same problems of a theistic God.
Proof requires definitions that are objective, otherwise it is just subjective experience which cannot be objectively proven. God is beyond being objectively defined. God is supposedly the 'Self-Existent One', 'All Father', the 'Alpha and the Omega', 'The Supreme Being'. In addition to this God is also to be Omnipotent, Omniscient, and Omnipresent: that is all-powerful, all-knowing, and present everywhere at all times. The Bible says in the Book of Job 11:7-8: Can you find out the deep things of God? Can you find out the limit of the Almighty? It is higher than heaven - what can you do? Deeper than Sheol [place of the dead] - what can you know? Equally, in the Quran 43:84 And He it is Who in the heaven is God, and in the earth God. He is the Wise, the Knower.
Each of these definitions are non-definitions in the objective sense for any entity. Epistemology has abstractions that are related to concretes to make the concept that is the abstraction: e.g. various types of apples to create the concept of apple, and various types of fruit such as apples, oranges, bananas, strawberries and the like, to come to the concept of fruit. Without being tied to anything that is objective what remains is subjective interpretation of an external entity. There is a difference between the subjective experience of an external source, such as do we see the same shade of red as one another; that perception may differ. Variation of interpretation of an existential object's existence is in how we see a thing, not in what it is: it is either there or it is not. Whether one sees it or not does not have an effect upon its actuality. If we do not see the table in the dark, when we kick it our senses will alert us to its actuality.
When the books one bases one's understanding of God upon make the same admonition that God is beyond human comprehension, and with only disconnected definitions, what makes something objectively provable is impossible. To define something is to limit it to the definition, but 'Self-Existent One', 'All Father', the 'Alpha and the Omega', 'The Supreme Being' as well as being Omnipotent, Omniscient, and Omnipresent are each impossible to individually tie to a single entity, let alone combine all of them into one entity.
What is generally advanced as proof of God are 'experiences of God', meaning individually feeling the presence of God in one's life; they 'felt' the presence of God. This experience is not objective, but subjective, meaning it is beyond being verifiable to an actual being. There can be no debate that one may have felt some thing, but that one may have felt something has no bearing on the actuality of what is real in the world. Again, the interpretation of something has no effect upon what that thing actually may be. 'Feel the presence' of something, and is that something the Abrahamic God, Zeus, a goblin spirit on Mars, neural stimulation/activation? The first three each have the same potential proofs; the last one actually can be seen, but it is not the one that refers to an entity.
Non-personal experience proofs of God that are advanced are substitutes for great cosmological forces and events: the origin of the universe, the creation/existence of life. However, these proofs of God are interchangeable with the Deistic and Pantheistic concept of God making the willing God unneeded or irrelevant; Nature is the dominating and guiding factor, with God as a poetic image working with Nature. There is no room for a God that wills; Nature, with science being a good tool to observe, understand and predict events to varying degrees that continue to be fine-tuned, shows us that water when reaching a low enough temperature will freeze, when reaching a high enough temperature will turn to a gas, explain why lightning strikes and what causes and the effects of neural stimulation. This Nature's God is the Law of Identity: things being in accordance to what they are, and interacting in accordance to how they must. There is no God who wills; there is Identity in Nature of 'Is', that is A = A.
Each example of why it is impossible to existentially prove the existence of God, is also a reason of why it is impossible to prove that God does not exist. Without a specific object of reference, there is not a specific object to disprove. In a dialogue on such a subject between Arenos and Madgo, Madgo reprimands "You speak so highly of proof, then I ask you to prove that God does not exist."
"I tell you it is not for me to prove the nonexistence of something; it is for you to prove the existence of a thing. Here" Arenos turned his empty hand so his palm faced up. Looking at Madgo, Arenos continued "I say there is a rock in my hand; prove there is no rock in my hand."
"Don't be ridiculous. Of course there is no rock in your hand."
"How can I prove what is obvious?"
No proof is proof enough of the nonexistence of a thing. When proof is found, then there can be said for a thing to be. However, proof, or 'hints' at the existence of a thing unbeknownst to man does not mean that it is the willing God. A mystery by default does not mean the answer is God. Eclipses used to be seen as ominous, omens from angry gods or an angry God; however, with science, the supernatural veneer has been lifted and humankind has seen far beyond what any early man could have ever imagined. The Sun used to be seen as an object that moved around in our skies; we now know that it is one but of billions of stars in our own galaxy, and a small star in that. The changing properties of water, as well as the fields of biology, neurology, astrophysics and countless other areas of science show that there is no reason to believe in the God that wills. When there is no proof to first make an assertion of the existence, what more can be offered? Even if there was some proof of the existence of some thing, that does not mean it must be the mysterious, default vague notion of a willing God.
Finally, there are those who state that as there is no proof that there is no God, that is reason enough to believe in God for God will show Himself in His time, to those who are willing and able to receive His message. This is the last speculative resort of one clinging to what is beloved. The depth of devotion though does not have an effect upon the actuality of existence. In conclusion, I offer Bertrand Russell's 'teapot':
"If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is an intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense. If, however, the existence of such a teapot were affirmed in ancient books, taught as the sacred truth every Sunday, and instilled into the minds of children at school, hesitation to believe in its existence would become a mark of eccentricity and entitle the doubter to the attentions of the psychiatrist in an enlightened age or of the Inquisitor in an earlier time.”