MIAMI—Jeanette Morreal backed Republican Sen. Marco Rubio of Florida in his successful U.S. Senate run in 2010, but turned against him for his work on an immigration-overhaul bill three years later.

Now, the 76-year-old retiree in Cape Coral, Fla., supports New York businessman Donald Trump in the GOP presidential race, drawn to his strong stance against illegal immigration and his ability to get things done.

“He...