Update:
The HOA president wanted to meet with us last night and discuss the issue. I explained what the problems were and why we had such an issue with it. I explained that we measured and found where we believe the property line was as well as the previous conversation we had back in April about not drilling into our fence. We went back and forth for a bit, so I'll save you from all of that.
In the end, the HOA president seem to side with me that I had every right to be upset. He suggested we go ahead and get it surveyed to figure out exactly where the line is. He was going to also check on the easement rules to see about blocking off the access to the boxes. He then said that until this is settled, my neighbor should take down the gate and the easement fencing immediately instead of waiting for this to get figured out. I said that sounds more than fair, but he tried to put up a fight because he was concerned about their dog getting to run around. His wife even told him to take it down right now since the dog can live on it's leash for a while longer. Came home today from work and the fencing is gone.
Overall, it went very well and I think he finally got the idea that he's not going to win. Working on finding someone to properly survey and even pin the lines so we can get this wrapped up. I'm open to recommendations if there's any out there.