It would probably take some kind of objectively verifiable evidence.
Let's say that I was out for a stroll and suddenly a ball of light appeared before me glowing and speaking with a booming voice and talking about how it was a god and it had such and such power and wanted to contact mortals. Well that's all well a good, a personal experience would be nice, but it wouldn't be objective, I may well be hallucinating.
But if, as I was talking to the orb, people around me began to gather and they all saw it as well, and then the next day there were photos of it in on the newspaper that would be even better. Better still if the orb hung around town until scientists arrived to study it.
Even than though there's no guarantee we're dealing with a God. A sufficiently advanced alien culture might have technology that seems supernatural or that is designed specifically to deceive. Even terrestrial technology we currently possess is capable of some incredible things. So jumping to the conclusion that we are dealing with a deity isn't a good thing.
And than there's the question of whether such a being wants or even deserves worship. A God would have to be pretty full of itself to want mortals bowing before it.
Scientific evidence would be the quickest way to convince me.
To answer your first question last, there are quite a few reasons why I'm a non-believer. I have a hub about my upbringing and deconversion from Christianity if you're interested in the long-story. The main reason is because I sought the truth, I went looking for the "true" God of the universe after Christianity turned out to be complete BS and that eventually led me to realize that there was no good reason to believe in any deities.
The reason I REMAIN an non-believer is that no version of theism has ever met it's burden of proof. Every argument I hear for god's existence is logically broken or flawed and there is no solid evidence of any kind for any god's existence.