The Formula One world champion, father John, manager Richard Goddard and trainer Mike Collier escaped unharmed following the attack that occurred as they returned to their hotel in San Paulo after qualifying for the Brazilian Grand Prix on Saturday.
Button was ready to race yesterday but he described the incident as "horrendous".
"You hear about it happening over the years, but until you are actually involved you don't know how it feels, and it's a pretty scary situation," he said.
Button's father John, fears his son may have been a deliberate target. "We're aware of the dangers. We've heard of many, many incidents," he said.
"But the only thing that really worries me this morning is whether they were after Jense.
"It was really weird because we were in a very small, compact Mercedes, and they couldn't see in. But it looked like they were waiting for someone. I don't know if someone at the circuit had told them we had left.
"So that's a little bit worrying. Maybe I'm wrong, but yes, it was scary.
"Our driver, though, was awesome – straight out of Hollywood, the damage he caused."
The quartet managed to flee as their police protector, Daniel Toni, used the armoured Mercedes B-class car he was driving to ram his way past several vehicles.
A group of mechanics from the Swiss Sauber team was held up at gunpoint on Saturday and forced to hand over belongings.
Also some photographers have had their equipment stolen.