I've just read through the thread then realised it's a couple of weeks old - has the chicken problem been resolved yet?
I have chickens that free-range all day and I also have vegetable gardens. Here are some things that might work:
Observe how the chickens are getting over the fence. Do they fly right over or hop onto the top then flutter down? If you were to put an 'extension' of bird netting on top of the fence, this might be enough to deter them. By bird netting I mean the soft nylon netting just draped loosely between bamboo poles. Soft, loose netting seems to deter my chickens more than proper chicken mesh - they can't perch on it and hate getting their feet tangled in it.
Also consider loose bird netting over your garden beds, propped up with bamboo sticks where needed. The chickens will still be able to push in and get to things like strawberries that are close to the edge, but you could protect most of your garden that way.
I recommend black netting - it's virtually invisible so it doesn't look like your garden is a fortress. Around here it's really cheap to buy.
I think it's terrible that you should have to take action to keep your neighbour's animals out of your garden.
If you have to do something radical (eg trapping and removing the chickens) then the neighbours will just have to accept that they had the chance to solve the problem but left it all up to you, and you did what you had to do.
P.S. Personally I wouldn't keep the chickens. You could put a lot of time, effort and money into housing and feeding them properly, get them laying plenty of eggs, then find your neighbours accusing you of theft and wanting them back. :blink: