I've been lucky this year and had no problems with squirrels in my screech owl nest box. Significantly, I think I know why.
Upper-left: Screech owl nest box in gray.
Middle: Fox squirrel nest box in green.
Though I've seen fox squirrels taking an interest in my squirrel nest box at various times, for lack of built-in video cameras or other instrumentation, I remained unclear as to what, if anything, was going on in there. However, lately, I've seen a good sized female relaxing on the horizontal board that serves the double-purpose of being a stand-off from the tree limb that keeps the nest box hanging vertically from a non-vertical limb, and as a walkway from the tree trunk to the entry hole in the bottom of the box. Around these parts, it's the right time of year for a fox squirrel to have her winter pups (well fed, as I expect my bird feeder ensures, a fox squirrel can produce three litters a year). Taking the afternoon air on that horizontal board, Mme. Squirrel could easily listen to her pups’ activity in the nest, while keeping an eye on everything going on around her, and from a position of complete safety, so it’s a good arrangement for her.
It’s gratifying to see the squirrel box serving its intended purpose, but I’ve also come to a reasonable degree of confidence that it’s serving an additional role: keeping other squirrels out of the screech owl nest box. The way it does this is very simple: fox squirrel territoriality. While they may tolerate each other to varying degrees in foraging encounters, they do not seem willing to allow other fox squirrels to setup a nest in the tree that holds their own nest. (Perhaps in really enormous trees there’s room for “my side; your side”-style compromise, but around here, even when a tree contains multiple squirrel nests, they all seem to be property of the same squirrel or matriarchal family group.) So, having an established squirrel nest more-or-less next door to the screech owl nest box ensures that it has a squirrel preventing all other squirrels from moving in, and you could not ask for a better, or more devoted, defense than squirrel self-interest.
So, if you've been having a problem with squirrels in your screech owl nest box, perhaps building and installing a squirrel nest box (similar dimensions to a screech owl box, but with an entry hole in a rear corner of the floor, and, perhaps, multiple internal levels) nearby is worth trying. And please report your results back here.