It's gonna be awfully tough to get a coyote into a box trap, and a box trap is really the only way that your neighbor's dogs won't get hurt.
Set up like you're calling them in the daytime. You already know where they hang out, so mount a a bright and inexpensive light (Surefire G2) to your rifle or shotgun, put a red filter on it and use some puppy distress, coyote distress, or good ol' prey distress and wait for a coyote to show up. Scan the area slowly with the red light, kill them when they show up. And I recommend shooting with your setup a few times before actually using it for hunting - if you've never done it before, it can be a little awkward at first.
And why are you restricting yourself to night hunting anyways? Call them morning, noon, and night - you'll thin more out that way, and you can hit them when they are at their various hangouts throughout the day.