Can You Prove God Doesn't Exist?
One of the reasons why I don't think people should ask theists to prove their god exists is due to the likelihood the response will be, "Can you prove God doesn't exist?" This is a problematic question, not because it's difficult to answer, but because, if a person responds this way, you can be sure that any answer you give will require a great deal of further explanation. Of course, you may find yourself being asked this question by a theist offensively, in which case, here's the route I think is most useful to take.
You may be tempted to be forthright, and say, "Yes, I can prove that your god doesn't exist," and then commence one of the several proofs against the existence of gods, but I don't suggest this approach. Most proofs against the existence of a god only work on specific definitions of gods, and it's presumptuous to assume you know their god well enough to disprove it. Also, most people don't really understand how logical proofs work, and a fully sound syllogism is useless if people can't understand it. It's also possible you've encountered a theist who knows a little bit about logic, in which case you'll find that their definition of a god is so contrived it's immune to all the usual proofs.
It's also tempting to turn it right back on them and ask, "Can you prove unicorns don't exist?" or, "Do you believe in things just because they can't be disproved?" and if you're in a public debate, this may be a good tactic, but in one-on-one conversations, this will most likely backfire. It's hard to say which they will find more insulting, putting their god on equal footing with a mythological beast or suggesting they'll carelessly believe anything; either way it's pretty much impossible to convince someone of anything if they feel they're being insulted.
Theists become believers, not because they can't disprove the existence of their god, but because they're convinced by teleological arguments (everything looks designed), cosmological arguments (everything had to come from something), and personal experiences. They may not know the formal names for these arguments, but they know the general ideas, and they find them compelling. From the theist's perspective, the burden of proof is on the minority atheist.
You could try pointing out the flaws in their evidence, but that's an uphill battle. Yes, teleological arguments are negated by evolution, but they will likely think evolution is evil. Yes cosmological arguments are childish in the light of physics, but this requires an understanding of relativity and quantum mechanics. Yes, personal experiences are often the result of delusions, confirmation bias, group think, selection bias, and the like, but using these terms comes across as highly rude. I think a better approach is to take their question to task by saying, "That's a good question, what sort of proof do you think it would take to prove that a god doesn't exist?" The reason I prefer this approach is because it moves the talk from the realm of argument to the realm of discussion which is far less confrontational.
It is highly unlikely that they will have a list of criteria prepared, and much more likely that they will say, "You can't prove that God doesn't exist." This is when I think it's safe to point out that believers of different religions might say something like, "You can't prove that Zeus doesn't exist," since neither of you believes in Zeus. Then you can wonder aloud at how the two of you might go about arguing that Zeus doesn't exist. I like to start abstract and say, "Despite what some people think, you can prove a negative." For example, we can prove a square circle doesn't exist because a square must have straight sides, while a circle must be curved, therefore, a square circle is a logical contradiction. We can apply this reasoning to certain types of gods as well. Have them suppose a god that is described as, "always perfectly good and always perfectly evil." We can prove that such a god doesn't exist because it too is a logical contradiction; nothing can be both perfectly good and perfectly evil at the same time. To continue with Zeus, you could mention that the ancient Greeks believed that Zeus was the cause of lightning, but we now know that lightning occurs naturally. However, just because we can demonstrate that lightning doesn't come from Zeus, it still doesn't prove he doesn't exist, in fact, there probably isn't anyway to disprove Zeus, but we can show that aspects of him are unnecessary.
I usually like to end at this point and trust that they will connect the dots without my help. This way, I can introduce them to the basics of logical contradictions, show them that being unable to disprove something is not a good reason to believe it exists, and hopefully allow them to be okay with the idea that non-evidential beliefs can be replaced with evidence-based beliefs, all without ever alienating them by directly attacking any of their own beliefs.