Maybe things have changed, but their current policy is no

http://www.fedex.com/us/freight/rulestariff/prohibited_articles.html

Firearms
Carrier will transport and deliver firearms as defined by the United States Gun Control Act of 1968, between areas served in the U.S., but only between:
Licensed importers; licensed manufacturers; licensed dealers; licensed collectors; law enforcement agencies of the U.S. or any department or agency thereof; and law enforcement agencies of any state or any department, agency or political subdivisions thereof; or
Where not prohibited by local, state and federal law, from individuals to licensed importers, licensed manufacturers or licensed dealers (and return of same).

Yes, that's according to the shippers' policies, not the law.

I don't think UPS and FedEX are the only commercial carriers you can use.  After all, it's legal to include firearms in a household goods shipment when moving.  I guess one would need to shop around for a carrier if UPS or FedEX refuses your package.

Doing some forum reading, it seems UPS and FedEX are inconsistent in their enforcement of their policy.  Based on the policy you posted, you'd have to use a common carrier and address the handgun to an FFL at your destination, then do the same if returning.

 

Looks like UPS is okay with shipping from or to an individual (non-FFL) if you are a collector (C&R 03-FFL)....

Special Procedures for Shipping Firearms

UPS accepts packages containing firearms (as defined by Title 18, Chapter 44, and Title 26, Chapter 53 of the United States Code)
for transportation only (a) between licensed importers, licensed manufacturers, licensed dealers, and licensed collectors
(as defined in Title 18, Chapter 44 of the United States Code), and government agencies and (b) where not otherwise prohibited by
 federal, state or local law (i) from an individual to a licensed importer, licensed manufacturer, licensed dealer, or licensed collector;
and (ii) from a licensed importer, licensed manufacturer, licensed dealer, or licensed collector to an individual.

https://www.ups.com/us/en/help-center/packaging-and-supplies/special-care-shipments/firearms.page

Based on that, if you have a friend or family member with a C&R FFL, UPS is your friend.