"I'm not feeling well" is a good set phrase you can use when you want to tell someone that you feel sick, but you don't want to specifically say what's wrong. "I'm not feeling well" can be used when you have a headache, stomachache, a cold, diarrhea, menstrual cramps, or other health problems. You probably wouldn't use it if you broke your leg, had a heart attack, or had another major health problem.
It's common to say "I'm not feeling well" as an excuse for not going to work. Your boss might believe you, or might not. But most bosses won't ask you more specifically why you need to stay home because that information is considered to be very personal.