'God does not exist' is what is known (in some forms of the scientific method) as the NULL Hypothesis. A null hypothesis is the positive TESTABLE assertion that some entity or phenomenon DOES NOT EXIST.
Examples: Black swans do not exist
Bigfoot does not exist
Atheists do not exist
The null is always paired with an ALTERNATIVE Hypothesis, that the same entity or phenomenon DOES EXIST.
Examples: Black swans do exist
Bigfoot does not exist
Atheists do exist
In this kind of scientific method, the scientist then makes a prediction about what kind of evidence could be collected that would, if absent, would support the hypothesis that the entity or phenomenon does not exist, while if the evidence were present, would support the alternative hypothesis, that the entity or phenomenon does exist.
Europeans long thought that black swans did not exist, because all the swans they knew of were white: they collected information and found no reason to reject the null hypothesis and accept the alternative. But when Europeans got to Australia, they discovered that black swans do exist, which caused them to reject the null and accept the alternative.
So far, there is little accepted evidence that Bigfoot exists, so scientists 'fail to reject' the null hypothesis.
Likewise, there is considerable evidence that people who do not believe that god exists (aka atheists) exist, and so the null is rejected and the alternative is accepted.
The statement that God does not exist is a null hypothesis.
The alternative hypothesis is that God does exist.
The point of contention is over what would constitute valid and testable evidence, the absence of which would cause one to fail to reject the null, and the presence of which would cause one to accept that the alternative is true.
The real problem is that there is no clear, testable definition of what God is, or what evidence would be collectable, in order to reject either of the hypotheses.