Trying to identify brakes

Would the new calipers work if I swapped sides to have the caliper mount to the rear?

Yes. The calipers are left and right, and you'll want to switch them side to side to keep the bleeder screws on the top.
Cardone 184065 & 184066 are $40.99 on Rock Auto. That includes core charge, but not shipping or tax. There's always a 5% discount code. Search this site or Google for it.
These calipers use screw in hoses, not the square end with the banjo bolt.