Yes, it is correct to use "ill at ease" in the sentence as you did. The English phrase "ill at ease" means uncomfortable or uneasy. Thus, your example means "I feel uncomfortable when someone stares at me. Other examples: I feel ill at ease accepting a gift from a stranger. I often feel ill at ease when I am asked to speak in public.

ill at ease=worried and uncomfortable; uneasy; anxious "The boy obviously felt ill at ease about the interview!" I think you example is good!