What's special about the 1 wing?
Is it just for First-Class SOLDIERs?
How do they get it?

I could also tell you about Ultimecia's actual Powers, but another day.
Squall can defeat Ultimecia by his own if you choose to do so.
Except Squall doesn't rely on mako at all and can un-junction his GFs while SOLDIERs are infused by (and stucked with) Mako.