There is one solution that avoids your cats needing collars - get one of the locking cat flaps which operates from their microchips (and I do hope yours are all chipped, it's the only way we ever got our Rover back when he went missing, do it now if you haven't already). You programme the catflap to each cat's microchip, so only they can get in and out - well worth it, in your case, IMO. You can still lock it manually if you need to keep them in (our out).
It won't stop neighbour-cat lying outside the house and mugging your cats when they go outside, but it WILL stop him being able to get in.
I would never go back to collar operated cat-flaps - the magnets are chunky and collars for cats are dangerous, Rover nearly hung himself twice with his ex-collar.
You can't blame your neighbours for their cat's behaviour - you need to make your home anti-visitor, they won't be able to keep him out of your house.