I had my navel pierced for 10 years is it normal to have puss and blood come out from my navel below the piercing when I get sick with an infection?

Of course not normal. This is definitely abnormal. It sounds like you have a chronic infection at the site, or perhaps a sebaceous or inclusion cyst. Not sure what you mean by symptoms (only?) at times of (other?) infections. What kind? Colds? Maybe just a coincidence. This could get severe in the future, maybe need minor surgery to correct it. See a doctor promptly the next time this occurs.