Should I use flash outside on a sunny day?
YES!!!!!
However, I always thought the answer was no. I assummed on a bright, sunny day there was PLENTY of light and no flash was needed. What I didn’t take into account were all of the shadows that were cast by this bright beam of light! 
Just the other week at Thankgivings I was taking some shots of my boyfriend. He refused to remove his cap because of a fresh booboo he’d just gotten on his forehead. So I thought, maybe the shadow from his cap won’t be too dark. Man, was I wrong. Here’s my picture to prove it:
Harsh Shadows on Face in Bright Sunlight
Can we say LESSON LEARNED!?
1) He’s gotta take off the cap!
AND
2) I gotta use flash on those shadows!
All I’m telling you here is to turn ON your flash if you see harsh shadows on your subjects face in bright daylight. 
I tried it out myself and here’s what I got.
Portrait in Bright Sunlight with No Flash
Portrait in Bright Sunlight With Flash
As you can see, my face is much better illuminated and the very harsh shadow from the sun has been dimmed by the use of the built-in flash on my camera. Nevertheless, the sun was just too bright in my face and it was difficult for me to hold my eyes open. I perhaps could have turned a little more away from the sun to prevent this.
FLASH TIP:
You will probably have to force the flash to fire in such bright light. You can do this manually by popping out the flash on some cameras or you can go into your cameras menu to turn on the flash in the settings. Also, you could select the night portrait scene mode.
Getting Creative
You may want to get creative and bring out harsh shadows. If that’s the case, go on with your bad self! Leave the flash off and have fun. Partially lit faces can be quite mysterious.
Learn More about Fill Flash
Here are a few good pages for advanced learning on how to use fill flash outdoors: