Can I use Toyota coolant/antifreeze on my Honda Civic if I were to completely flush it before doing so?

I know it sounds weird but the reason I'm doing this is because I work at a Toyota dealership and I can get the fluid for half price

It's all ethylene glycol and the correct mixture is 50/50 for all vehicles. There isn't a bit of difference from one make to the other.

You could use a universal Engine Coolant but not one specified for use in a different car

Yes you can as long as you do a complete flush. Hondas and Toyotas usually have aluminum blocks so the A/F should work just fine, after all the stuffs only got 1 job, and it's to keep the engine cool.

These things must be done delicately. There are more than a dozen different chemistries; putting Toyota coolant in a VW with existing Pentofrost can destroy the engine in minutes as the coolant gels. Be very careful about choice of antifreeze.

Toyota coolant is HOAT, zero amines, borates, silicates, phosphates, and nitrites. Honda dark green coolant is OAT but otherwise compatible. I personally would not use the Toyota coolant in a Civic. It is important enough to me that I will not buy a used car with the wrong coolant in it - I learned that lesson the hard way.