Can a US president use a personally owned jet with the call sign AF1? Would the Air Force be obligated to add security and defense measure to this aircraft if the president refused to fly in the VC-25?
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.