Burning pain in right buttock, accompanied by burning that picks back up a few inches below my knee and runs down the outside of my leg, through the heel. Presents only with sitting. Possible causes?

Piriformis syndrome. Maybe a "peripheral sciatica", and the nerve could be compressed by the piriformis muscle. Alternatively, could possess focal sacro-iliac dysfunction, and perhaps best source for relief of both from a visit to osteopathic specialist.
Possibly a sciatica. A sciatica is caused by an irritation of a nerve in the back that then can cause pain that runs down the leg like you are describing. There are also potentially other causes as well. Usually a detailed history and exam can determine the cause. Thank you for your question.