Unless you're using a hammer to tap it out or a drill, chances are nothing will get damaged if you do it once. If you make it a habit to clean your guns incorrectly, then there may be damage over time especially around the crown or grooves. One of the most common reasons a fire arm gets damaged is due to incorrect cleaning. If you correctly clean your fire arm, you could clean it 100 times a day without any issues.
When it comes to barrels, it is far better to "pull" than to "push" something through. This is especially true with tighter barrels like a .223. If you're especially paranoid get a pull wire or use a guide when cleaning. If you insert the long piece of rod into the barrel first, then pull the brush, it will be very easy to clean compared to trying to push it through.