Although I think both your neighbor and the Board of Health are overreacting, won't they be happy if you simply remove the squirrel feeder? It doesn't sound to me as if you stand to "lose it all." Some battles just aren't worth fighting. Furthermore, I bet you can continue to feed the squirrels, but more surreptitiously, by not using a feeder.
For what it's worth, squirrel feeders are a primary point of transmission for fungal infections that cause hair loss and other communicable diseases that affect many squirrels, so you might actually be doing the squirrels a favor by not causing them to concentrate at your feeder.