How do you get a Jack Russell to shut up?

Sounds like you might have and authority problem. If the dog knows certain command such as sit then just  sit him down in front of you and tell him "no". If the dog stops barking great if not continue to say "no". If this is not having any effect either. Lay him down on his side and put you hand at his neck, apply only enough pressure to keep the dogs head on the floor. For dog this method is taking the alpha position of the pack which you should already have. After 15 seconds let the dog up, if he backs more say "no" and put him back in the floor/neck position for another 15 seconds. Patience is one thing you will need sometimes it will work over night other times it will not.

Being alpha lead make you the boss of three dog and that will make it easier for you to teach it to sit, lay down, stay, heel ...

Hope this helps

Let it know that it's not right to bark. Spray it with water, like with a spray bottle, or buy a muzzle and put it on him when he starts barking too much. You have to be paitient but eventually he will learn that he gets punished for barking, and won't do it.

If your Jack russel does not obey look it in the eye and say 'No' and repeate it in a loud tone and soon he will learn. If you want it to go in a cage say to it 'go' and point to the cage and if he goes give him a treat while he is still in the cage. Because he will not know why you rewarded him for doing out of the cage.
P.S I am not a pro I just learn't it today