Can a U.S. President have a non-government owned aircraft that would routinely be given the call sign AF1?
Air Force One is the call sign of whatever USAF aircraft (see @slookabill's comment below) the US President travels in. Once he boards another aircraft, the call sign AF1 is transferred to that aircraft (for a dramatic example of this, see the climax of Air Force One movie).

If a president were to fly in their own aircraft (or commercially) the appropriate ATC call sign would be Executive One.



As far as fitting out a personal aircraft, the VC-25s are USAF assets (operated by 89 AW) and are highly customised and specifically configured for the job of transporting the president (and support staff), costing over $300 million. I doubt any president is going to get the option of flying their favorite aircraft on a regular basis instead of the state aircraft.

It is not likely that the government would pay to configure another (non-USAF) aircraft for the president to the same specifications as the VC-25s serving as Air Force One. After all no US President's personal car has been configured as the Presidential State Car so far.