When you say "fear of being sick", are you talking about fear of vomiting? Emetophobia is the fear of vomiting, and it is more common than you think.
Anxiety sufferers usually each have their own specific fixation of worry, whether it be vomiting, dying, going crazy, having a life threatening medical condition/illness, etc. There are many fears like that that people with anxiety (usually more specifically panic disorder) struggle with.