(not an REI employee)
I'm 99.99% certain, yes, will not damage tent (not talking about the poles of course)
consider just stuffing tent & sleeping bags (if tent is dry), into the bottom of pack, which then fills up all the 'nooks and crannies'.
a compression sack, becomes a 'bowling ball' in your pack, and it's tough to use the 'wasted space'.
@lilithwantsu in agreement with @Philreedshikes - yes, you can certainly carry the REI Trail Hut in a compression sack, likely not including the poles. That said, we wouldn't recommend storing it for long periods of time compressed - and you certainly don't want to store it wet. Hope this helps!
Not an REI employee but...assuming this is for backpacking...
Carried...sure but be careful of any sewn in struts for things like vents since a compression sack may distort of even break them... You might get some compression if you stuff your tent. If you fold it you might not get much. It depends what you trying to achieve.
Stored...maybe not so much...a looser stuff is probably better for that.
...but as Phil says loosely stuffing a dry tent in a pack often works better. You do have to be a bit more careful with sharp things if you don't protect the tent with its own stuff sack.
One method is to use a compactor bag as a pack liner and stuff all the things that must be kept dry into that...then twist, and roll the opening to cinch it. Then put all the things that don't have to be kept dry or are already wet on top. With a two walled tent you might try to keep the body dry if it is but keep the fly handy so you can dry it. You still want to try to keep heavier stuff close to your back if possible...generally that is food unless you are fond of traveling with your lead weight collection.