You are right: the burden of proof rests upon the people making the claim. It doesn't matter if it a affirmative or negative claim. It doesn't matter if the claim is universal (all) or particular (some). If you make a claim, you should provide evidence for it.
It is impossible to prove or disprove many things in this world. Ask any philosopher. We can, however, provide evidence for our claims. That evidence does not always prove that our claims are true, but they do prove that our claims are not necessarily false.
If I told you I am very rich, I can support that claim by showing you my expensive watch or Italian sports car. That doesn't prove I am rich (I could have borrowed or stolen these), but it does prove that I am not necessarily lying.
Christians are not interested in proving or disproving the existence of other gods. If we went around disproving other gods, we would be kept busy until the end of time. We believe the Biblical claim that God is one, and that the Trinity is one God in three persons. We provide evidence for this, but we cannot *prove* it. Therefore, we believe that the existence of other gods contradicts the Bible, and are therefore false.
Atheists do not provide ANY support for the non-existence of God. In fact, they disregard or overlook all evidence that provides support for the claim that God exists. They commit the negative proof fallacy, saying since we cannot prove that God exists, that means God does not exist.
While there are many lines of reasoning that support the existence of a Supreme Being, the identity and nature of that Supreme Being involves an entirely new set of arguments. I can lend support for my claim that the God of the Bible is the Supreme Being, but when discussing theology with atheists, it is difficult to prove to them that any Supreme Beings exist.
Furthermore, atheists commonly criticize and attack the Christian (sometimes Muslim) God. When a Christian presents arguments for their belief in God, atheists will then question which God really exists. They commonly equivocate this way in order to avoid losing the debate.