Is it unhealthy to workout everyday?

No way! Exercise is wonderful for the body, heart, immune system, etc. We are finding out more and more about this everyday. Check with your doctor before new regimens or particularly stressful activity, however!
No. Daily exercise is generally a good idea, and not harmful. I have been doing so for about 30 years now. Of course, common sense applies in terms of how much you exercise and what you do. You should feel good during and after exercise for the most part. If it causing pain or other discomfort, consider backing down a bit and discussing it with your health care provider.