Could a wall be built on private property and be deeded to the government after it is constructed?
Theoretically, maybe, in the past, but I don't think much of the actual southern border land remains private.
The U.S. border is itself a federal designation; over the years the U.S. government has federalized (seized) much of the southern border property.
Texas land grab: How the federal government abused its power to seize property for the border fence (article from a year ago, written from an anti-wall perspective)