Day hunting of coyotes is better fun, especially in the snow, but night hunting can be more productive. Contrary to popular opinion, red lights can spook them on occasion, had it happen on a text book set up. But, other times they will stand still long enough to gain a good sight picture. If I were to muzzleload specifically for coyotes it would be a daytime proposition only. For night hunting a light mounted on a scope with a pressure switch taped on the forestock is the most practical.
You don't seem to be taking this up for sport but to put the pet eating pack on notice. May as well set yourself up for success. Coyotes are smart and educate quick. One calling session doesn't do it. You have to have multiple setups predetermined and be willing to call in one spot for half an hour, then hop to the next and do the same. If you miss the first time they may, or may not come into the call the second time. Good coyote callers spend an exorbitant amount of time in the field. It was one of the most rewarding hobbies I've participated in before the market crash of 2008 and gas prices spiked to $4.00 per gallon.
Really serious eradication measures include traps and snares.
Be careful, it can be addicting.