You can install the 6.8 SPC upper on the 5.56 lower receiver. The lower receivers marked 5.56, 6.8 SPC and .450 Bushmaster are all the same except for the caliber markings. When complete 5.56, 6.8 SPC or .450 rifles are manufactured the lower shipped with it is marked with that caliber.
You should use the 6.8 SPC magazine that comes with the 6.8 SPC upper assemblies.
Subject: 6.8 SPC in .223/5.56 magazine
The two magazines use different followers as the rounds are different diameter even though they are the same overall length. If you load more than about 6 rounds of 6.8 SPC into a .223/5.56 aluminum magazine the body of the magazine swells so it is very difficult to insert into the receiver if you can and the rounds will not feed properly.
The 6.8 SPC magazines have stainless steel bodies that do not swell and the proper followers for reliable feeding.