As Kurt said, not officially. I think you may be able to hike in to the campground and camp though. We never camped at Spruces in the winter, but we used to winter camp up towards Dog and Desolation Lakes which are just down a little and on the opposite side of the road (and they include a hike in).
I don't know what the official word is. I haven't stayed up there this winter, but I did last year. You just need to hike in. Great snowshoeing and winter camping!
Looks like you have to get a permit: http://www.fs.usda.gov/detail/uwcnf/about-forest/districts/?cid=stelprdb5048559 "Campsite Selection - permit is only valid for Jordan Pines or Spruces Campgrounds in Big Cottonwood Canyon. Camping is allowed in signed reservation sites only. A map is available. Keep ski trails and roads clear. Please do not camp on roadways or trails."
We used to dig snow caves up there when I was a kid.. well really we just dug under the picnic tables but they made great snow caves with very little effort I also remember stopping to pick up our permit on the way there... Enjoy.. don't forget to go up to donut falls to check it out when it is frozen