Hearing that kind of stuff hurts like h3ll! At this point, you don't know if and how much of it is true. Right now it is just hear say...pure neighbor gossip.
If I was in your shoes, I'd probably go talk to the neighbor who is spreading the rumors and find out more info.
If the info that you get adds up and you think there is some truth to it, then you can decide what to do with it.
I would not confront your H right now. Wait to find out whether the info is credible. And if you choose to confront your H, then you should definitely think through what you want to get out of it and how you are going to approach the "confrontation".
I agree with BklynMom that it doesn't sound like your H would have had anything with xBFF, otherwise they'd likely be "together" now.
If this info happens to be true (which I don't think it is), then how does it affect your feelings for your H? Would you still want him back? Would you be willing to forgive him?
As much as it hurts to hear this stuff, it really is just gossip. It could have been you and your H and the neighbor just didn't recognize you. Or, it could have been friends that you had come over and the neighbor thought it was your H.
If your H would not have dropped the bomb and you heard this info, how would you be reacting? Would you think the neighbor was crazy or blind? We tend to think of the worst and believe when we think it sounds like it could be true.