Why do I get experience flushed skin on my neck and face every time I get sick?

Fever or rosacea. Many folks flush whenever they have a fever, so if you run a fever when you have a cold, it's as simple as that. If not, you probably have a tendency toward rosacea. Many things can trigger the skin flushing of rosacea, and illness is one of them. If the flushing is significant have a doctor examine you, because there are some rare, esoteric things that can cause flushing too.