The answer is generally yes. The rear leaf springs on the Cougar are longer, so nothing doing there. Also, the spring plates are different. Cougars use a wedge and rubber block arrangement (where the U-bolts attach), so the U-bolts are longer. I don't know about driveshaft lengths, but this may vary depending upon the application.
There is an excellent resource for interchanging parts.
Ford Parts Interchange Manual can be picked up at any Mustang vendor for $20 and is great.
It covers interchanging parts for all 59-70 Mustang, Fairlane, Torino and Mercury Cougar and Cyclone.