Last year we drove from the Natural Bridge to the Alto Vista Chapel stopping at the Gold Mine ruins and a few coves along the way. We did see a car out there and he was struggling to make it through a few low spots. He had to really pick a good line or he would have blown out a tire for sure.
I don't think the insurance from the rental agencies cover damage while off roading. If you take a car out there the chances of damaging it are far greater. Read the posts regarding insurance. There's some good information about how to make sure you are covered while offroading.
This year we're adding the California Lighthouse and the caves in the National Park to the trek. Should be fun. I really, really suggest the jeep though.
Aruba in 6 days!!