Because they'll not only not overturn it, but they'll create new precedent that will officially neuter whatever force the Commerce Clause has left (as little as that it may be).
My reasoning? Current precedent is already dangerously close to rendering the Commerce Clause totally futile. The Depression-era case, Wickard v. Filburn, and its progeny, most notably "Wickard for Pot Growers" (a.k.a. Raich), say that growing your own wheat (or weed) for your own use affects interstate commerce, even if you do it within a single state, and therefore Congress may regulate it under the Commerce Clause. The rationale is that when you grow your own wheat, it affects the interstate market for wheat (i.e. less demand from you). The same rationale was used for marijuana -- even though there is no legal interstate commerce in weed.
The same rationale will apply for the mandate. They'll say that one's use or lack of use of medical insurance and related services affects the interstate commerce in those markets, and that forcing us to purchase something is within the realm of appropriate regulation. By not having insurance, I subtract from demand, which should reduce prices (inasmuch as medical products and services are actually affected by supply and demand anymore).
But legally speaking, any person's existence requires consumption that "affects" commerce, and if that level of consumption qualifies as "interstate commerce," then what doesn't? The government might say that any "economic activity" falls under the clause. But then what's not economic? My choice to have sex with my wife instead of a prostitute affects how much money that prostitute spends. My choice to have kids certainly affects how much money I'll spend. Anything I do basically affects an interstate market. But if Congress can regulate anything, then the restrictive clause doesn't have any meaning!
In other words, the lawyers will ask the Court: Does simply existing (living and breathing and eating) affect Interstate Commerce for the purposes of the Constitutional check on Congress? And based on Wickard, the Court will be forced to say "Yes" unless they want to overturn a lot of decades-old precedent. If the Republicans push on this, I guarantee that they (and I, and eventually you) won't be happy with the result.