The Air Force VC-25's are Air Force One, Marine aircraft are Marine 1, a private jet is Executive 1. What about when the president is using one a private helicopter?
The callsign of a civil aircraft transporting the President of the USA is Executive One, as you already mentioned yourself. An aircraft owned by a corporation which happens to share the name of the president is not any different. (According to a comment by Dan Sheppard, Vice President Nelson Rockefeller used his personal plane, whose callsign was Executive Two – this would be the closest thing to the situation you ask about that has actually happened.)
However, as was already mentioned in another answer, it is highly unlikely in this day and age that the Secret Service would allow transporting the President on anything other than one of the VC-25s, VH-60Ns, SH-3s, or "The Beast", publicity stunts such as George W. Bushs flight with the Lockheed US-3A notwithstanding. Note that this was the only time so far that the callsign Navy One was used, which tells you something about how often something like this (POTUS using an aircraft other than one of the VC-25s, VH-60Ns, or SH-3s) happens.
Vanity Fair says that POTUS is not allowed to use his private helicopters due to safety protocols.
They will not have any special callsign when flying (since he won't be aboard), and they would most likely use their tail number, or something mundane like "Trump 1" and "Trump 2".