The answer to your question, Kim, really depends on the circumstances involved. For example, since mammals are warm-blooded, nibbling on an ice cube or two is not likely to have any impact on an animal's body temperature. This means that their bodies have ways to keep their temperature levels normal in most situations. But, if a pet is placed in an icy tub of water, is left outside in the cold for too long or is otherwise exposed to too much ice for an extended period of time, the body's temperature can become unstable and a dangerous drop in temperature, known as hypothermia, can occur.
The bottom line? Offering your beagle mix a few bits of ice here and there is not an issue. Just don't feed her huge amounts of ice or expose her to environmental situations that could cause her temperature to drop.