A question about time signatures in Styx songs.

Wow, every song?? I would contact Rick Beato on his YT channel. The guy is excellent at explaining song structures, one of the best out there.

Well, Come Sail Away is in 4/4 all the way through; there's a start for you! I'm not too familiar with Styx, but I've never really known them for experimenting with a lot of time signatures. It's a bit of a tall order to ask for an analysis of every single song within a certain time period - unless you're talking to a drummer in a Styx tribute band, I don't think many people are going to be that dedicated to the music of Styx to have an answer for you. You might get better results if you count along yourself and ask for help regarding specific songs that you're unsure of.

My guess is that Styx rarely wrote songs other than 4/4, though maybe the first Styx album (their proggiest)  could hold some surprises

If you're asking.... you can't hear it. If you can't hear it, you'll have problems playing it. ;-)

Seriously. Count it out. You can learn how to count time signatures or you can ask what signature every tune you're going to play is in. The former method is actually easier, in the long run. 

Styx has actually done a number of songs in time signatures other than 4/4. Starting with “A Day” on Styx II that starts in 4/4 and the first solo section (guitar duo)is in 6/8 then the keyboard solo is 5/8…
Fooling Yourself from The Grand Illusion starts with a strong 6/8 keyboard intro and the verses and chorus are in 4/4 but the keyboard solo is in 7/4, the end of the solo is in 4/4 and then it goes back into the chorus but the ending does the intro again in 6/8 then breaks into a strict 3/8 or 12/8.
Come Sail Away does stay in 4/4 during the surreal keyboard solo, but it’s not a typical 4/4 it’s more like a 3/4 3/4 2/4 feel which adds up to 2 measures of 4/4.
Sing for the Day is in 6/8 with some very tough keyboard work.
Pieces of 8, the song, sounds like it changes a lot but it really stays in 4/4, but they just change the feel of what the feel basically cutting it in half but if you tap the same beat from the start, it stays the whole time.
Cornerstone has an interesting intro with Borrowed Time, but it’s just accents that make it seem like it goes 3 3 2 which is a typical accent pattern.
Paradise Theatre: Snowblind is in 6/8… kind of a bluesy “2”.   Rocking the Paradise does have that 123 123 123 123 123 123 hit hit hit…. Hit booom whatcha doing…. But it’s still just 4/4.
Kilroy Was Here had “Haven’t We Been Here Before” in 3/4, rare for Styx
I don’t think Styx was ever after purposely creating songs in anything other than 4/4. I believe they just wrote what sounded good to them. Tommy Shaw had a tendency to like 6/8 even with his stuff after Styx’s main run through 1983z