"They had it when they were kids, they were miserable for a few days, and everything went well," Bevin said in an interview with WKCT, a Bowling Green radio station.
According to his election website, Bevin and his wife Glenna have nine children aged 5 to 16 years.
The governor says he supports parents who decide to vaccinate their children and those who reject it. But he said the decision should not be left to the government.
"This is America," he said. "The federal government should not impose this on people, they just should not."
CNN has spoken to Bevin to speak.
32 cases of chickenpox have been reported.
The teenager and his father claim to be discriminated against on religious grounds. Some Catholics worry about vaccines derived from abortion cell lines.
The Northern Kentucky Health Department argued that the ban was "an appropriate and necessary response to prevent further spread of this infectious disease." Chicken pox has not disappeared since the introduction of the vaccine in the United States in 1995, but each year more than 3.5 million cases are prevented by chickenpox vaccine, the CDC said.
Initially, only one vaccine dose was recommended for children. However, when experts found that a small proportion of children did not have robust immunity after the first dose, the recommendations were changed. Currently two doses are recommended: one at the age of 1 year and a second at the age of 4 years. The two-dose vaccine program has resulted in fewer outbreaks, according to the CDC.
CNN's Madison Park contributed to the report.