Funny this topic came up...usually, when you see a squirrel at night it is a flying squirrel. They are almost totally nocturnal. Gray and fox squirrels are not, but they will move, as posted previously, a little after dark or very early in the am just before the sun comes up. I have heard people talk about seeing them on a bright full moon night, but I still believe they are seeing a flying squirrel.
I have a cabin in the mountains of Virginia and for the last four years I have worn out three different brands of trail cameras watching the bears and bobcats...I have looked a literally tens of thousands of trail cam pictures, day and night, not ever seeing one single {1} gray or fox squirrel at night. Not one. I also bowhunt here at home and keep about 6 cameras set up in and around bow season...not a single solitary gray at night here either {big surprise} although we do seem to have a lot more flying squirrels...but even here, you might catch a photo of one once a month. They are real small.
I also happen to be a coon hunter...and that is where this thread is funny to me. Coon hunters often blame squirrels for causing their dogs to tree slick {a tree with no animal in it.} This is an embarrassing fault of the dog, nothing more...but to hear most coon hunters tell it, squirrels are out all day and all night long, they exist just to drive dogs crazy. In fact, squirrels never sleep. Most of them really know better, but don't dare get on a coon hunting forum and post that squirrels don't move at night...they will run you off there for ruining the best excuse they have for poor dog performance.