Even if Jack doesn't engage with a burglar (though I'm fairly certain he would if he was approached and grabbed), he's huge, has an extremely loud bark, is unafraid of circling people while barking, and has big, terrifying teeth. That would be enough to make me back off, and would definitely be enough to make a burglar back off. And even if that didn't work, he'd buy us the precious seconds needed to grab a firearm and increase the physical threat to the intruder.
Most burglars are cowards, though, and I think shooting them is unlikely most of the time. Even brandishing a firearm at them is enough to make them run away. They don't want to fight. They just want to come steal your stuff and get out ASAP.