So, if I buy a new bottom bracket, will I not be able to use the existing cups that are already there?

I thought I could just take the cups off, insert a new (square taper) spindle piece, then put the bearings/cups back on.
Essentially just turning the ends of the bottom bracket from cottered to square.


As far as the make/model, its a weird old little folding bike thats been repainted (not by me) and I really don't know much about its origins.
Just that its a single speed w/ a coaster brake.

Also, would it be possible to just take a used bottom bracket/crankset off an old 10 speed I've got lying around and just swap that out?

Thanks again for all the info!