Wow, there's a lot of mis-understanding of the theory of relativity going on here. Sunken, ignore everything everyone has been saying about not being able to turn on a flashlight at the speed of light, because length has shortened or time has stopped or whatever. It's simply not true.
Why? Because speed is relative. You aren't moving at the speed of light relative to, say, the box you're in. From your frame of reference, you'll never notice any change in your mass and what-not, because you never move relative to yourself.
If you were to move at the speed of light relative to some object, from your frame of reference IT is the one whose time, length, and mass are changing. Not yours.
Now, on to your question. Yes, you would be able to see the light coming out of the flashlight, and you would measure its speed to be the speed of light. This makes sense, because you know if you're driving in a car and you throw a rock out at 20m/s, you will measure its speed to be 20m/s, no matter how fast the car is traveling relative to the ground.
Now comes the tricky part. If someone were watching you whiz by in your box, and they had X-ray vision and thus witnessed you switch on the flashlight, they would also measure the light to be moving at the speed of light. The explanation for this is a bit complicated, and rather than explain it here, I'm just going to suggest that you read up on special relativity. I'll give you a couple good links to get started with below.