We sample a lot of box wines here, and sometimes even finish a complete box. Wine boxes are super easy to use, and the only tricky part comes as you approach the end of the box. Actually, of course, it's not the box that holds the wine - it's a plastic bag, usually mylar, that deflates as you pour the wine. There's no air in the bag to begin with, and no air gets in as you pour (unlike a bottle, which replaces the wine you pour with air). The fact that no significant amount of oxygen enters the bag is what lets box wines last for a month (or even more) after opening with little or no degradation of taste.
So, what happens as you get near the end? Several things... the bag inside the box may start to get crinkled up, and some wine may be trapped, unable to reach the spigot when you pour. You stand there, knowing that there's still wine, but nothing's coming out of the tap. Then, the spigot gives a little gasp, and the wine flows again. Good news? Actually, it's BAD news. That little burp relieved some of the crinkling in the bag, but it did so by allowing air into the bag. Once that happens, the process of wine degradation has begun. Avoid letting the box wine "burp"!
So how do you get the last bit of wine out of the box without introducing air into the bag? First, when you see the wine flow slowing to a trickle, do NOT keep pouring until it stops. A second later, it will suck in air. Instead, as you see the flow slowing, tilt the box to maintain a strong flow. This is usually effective for all but the very end of the wine. When even tilting isn't increasing the flow, and you think there's still wine in the bag, you may have to straighten out the bag to get the last few drops. Some boxes actually have an access panel in the bottom that lets you poke your fingers up into the box to try to squeeze out the last bit of wine. If this works for you, fine. Sometimes, just accepting the fact that the wine is almost gone and tearing the box open is the most expedient approach. It won't look pretty, but you'll easily be able to unkink the bag and get the last drops of wine out... with not a single burp!