I feel a pain under my big toe that feels like I have a splinter but I don't. What can it be? Someone mentioned it may be gout? Feet in good shape.

Tendonitis. If you're sure it's not a splinter it could be irritation of the fibrous connections under your toe. In my experience gout does often occur at the big joint of the big toe, but it is more severe than splinter pain. Take Aspirin or Acetaminophen (generic tylenol) and watch what happens. If pain persists or infection sets in, see a doctor. Soaking in plain water may help if it's tiny wood or glass.
Pain in big toe. Arthitis or malalignment of the joints of the big toe can manifest as pain on the plantar aspect. Please see a podiatrist and an x-ray might help diagnose the problem.