If we take breeding every year to be a sign of perfect health, then there are certainly some species that are in perfect health with only a temperature drop into the lower 60s F. From the way I look at this, these animals ARE "overwintered", even though they are kept indoors. They are getting the reduced temperature that they need to be fully healthy. Make sense?
On the other hand, there are species that will not breed unless they are taken to lower temperatures than what is typical indoors. Some people do this using a cold basement, or an unheated room. Some accomplish this by housing the animals in a garage or shed. Some use the refrigerator, but I'm not a big fan of this method.
I don't think it matters whether the animals you have were never previously exposed to a winter chill. You should get them cooler this coming winter. How MUCH cooler depends on what species we are talking about. The normal temp in your house may be sufficient.