Yes, they can and probably will someday for whatever reason more then likely go over the fence.
The downside to clipping their wings is it makes it harder for them to escape predators that might get under or over the fence.
At first read the neighbor sounds like a jerk, but in his defense chickens can and will do serious damage to plants. If he gardens he could lose hundreds of dollars worth of plants to a flock of chickens...I know it's happened to my yard...
I try not to think about it to be honest.
Secondly I know ppl who are petrified of birds. I don't get it personally but they do exist. Just like I am phobic about snakes. I realize it's stupid but I can't help it. I know someone who refuses to come to my home because of my birds inside and outside.
Then again maybe he just hates you. Neighbors can be like that.
Lastly, if he's talking about shooting your chickens I wouldn't wander over and try to bribe him with eggs. It might piss him off even more, make him think you're making light of his complaint for whatever reason.
I'd figure out how to contain my flock. Maybe put up a 6 ft privacy fence along that side of the fence....that way you could if needed put up an angled netting system to keep them inside better.
Or you could keep them in a run and only let them out to range when you are their to supervise.
Good luck !!
Julie