In truth, it's a must-win only for Wisconsin. Wisconsin is entering an extremely difficult 11-game stretch, perhaps even 12 games if you count at Northwestern as a test (personally I don't—not this year, without Crawford). As it stands, Wisconsin will probably be a significant underdog in 7 of those 11 games. Other than the home game against Iowa, this will be Wisconsin's best chance for a win. Which means they really need it if they want to make the tournament.
Illinois, on the other hand, has a home game against Northwestern and a road game against Nebraska coming up next. So even if they lose to Wisconsin and fall to 1-3, there's a great chance they'll right the ship and get back to 3-3 in short order. After that they'll have six very winnable games remaining and a decent shot to get to 9-9, making them a certain NCAA tournament qualifier. (Though I'd still predict 8-10—which probably still gets them in.)
In football, the more desperate team nearly always wins. This isn't necessarily true in college basketball. College basketball has a simpler formula: the home team almost always wins.
As you know, I think Illinois is overrated and most smart people agree with me. Though the polls have Illinois ranked as high as 12th, both Kenpom and the Sagarin predictor have them ranked 37th. And Illinois hasn't won a Big Ten road game in over a year. Most recently, they lost an ugly game at Purdue. But, to be fair they do have one of the season's most impressive road wins, at Gonzaga.
Wisconsin has looked far from invincible at home, where they've lost to mediocre Virginia and won a war of attrition against lowly Penn State. Wisconsin is ranked 17 in Kenpom and 25th in the predictor; the humans have Wisconsin in the land of no votes.
All told, a Wisconsin loss to Illinois would be surprising, but revealing—particularly if the Badgers come out and have another bad shooting night. If that happens, it will probably be time to resign ourselves to the fact that this team just isn't very good at shooting, and therefore not very good at basketball.
But I don't think that's going to happen. Stretches of bad shooting are common, at least for the Badgers. They have these stretches even when they are really good. For example, see the three-game stretch last year when they lost at home to Iowa and Michigan State, then on the road to Michigan: their effective field goal percentage in those three games was 37, 38, 38.2—each of which is worse than any their performances so far this year.
I expect Wisconsin to have an average shooting game and win. I will stick to my pre-season prediction of a 6-point win.