Any risk of damage/deforming tires by scrunching them up to fit in a box?
 

Some tires come squished up to fit in a really small package and are pretty flexible and others you get as is and they feel pretty resistant to bending. So with these tires that are hard to fold over, is there there any risk of damaging them if I fold them tightly? This is so I can ship in a small box.

Yeah, you don't want to kink the wire bead.

The difference is in the material in the bead. Steel wire is usually used on the cheaper tires and it is not good to kink that. The ones that come in the boxes folded are made of kevlar fiber instead, which won't take a kink or set. Nonetheless I still like to unroll them and let them relax out for a while before installing. I would not however put a wire bead tire into the same package size.
You can however put a Wire bead tire into a circular fold by twisting it over itself. This is how other wire things such as band saw blades are shipped in smaller packages. It is not as small as the kevlar tire package, but it won't harm it either.

Sheldon Brown (pbuh) has a small video showing how to fold a wire bead tire into three loops for transport. Remember to only use red toe straps.
Tire Folding