Do I cut small pieces of styrene to make a backing joint on each segment? Run a longer piece at the base/top to keep it square?
Yes. One of those. Or both. It kinda depends on what you've got around, honestly.
I'm a big fan of using square tube for such things, but bulkier strip stock works too.
Then again, I've actually got a building that I've had for over a decade and has appeared on a number of layouts that is held together with masking tape.