Hi, you might get more luck if you browsed the forums or started a thread with your question, but, short answer, no, theoretically at least it won't break out your skin. People turn to mineral makeup also because their sensitive skin is hard to please.
Zinc oxide, one of its main components is actually soothing, it doesn't contain any harsh chemicals from the regular makeup and it provides an SPF15 sunscreen.
Just make sure to read the labels as the drugstore stuff still has the same chemicals in it, and avoid bismuth oxychloride, potentially irritating.