I agree with the poster that your son might have some sensory issues.
My daughter has Sensory Processing Disorder. She was the opposite--any little tinkle was cause for us to change her diaper. When she was in daycare, the daycare lady told us one day to send in extra clothes--my daughter had expressed an interest in learning how to use the potty. She was trained (at 3 years) in one week, and never had a potty accident (peeing).
Going number two was a different matter; my daughter had lots of issues with that, including ecopresis. Her reasoning was the same as your son's for not wanting to go to the bathroom; she didn't want to take the time to sit on the potty and go.
For my daughter, it took going out and buying the Poop book, which shows all the different types of animals and their poop. It took her just long enough to get through the book that she spent enough time on the potty to get the sensation going so that she'd void.
Another thing a poster brought up: yes, having issues with bathroom "oops" can have to do with the other system. My daughter's inability to poop/not wanting to also caused her to have an overactive bladder, because when the body's constipated, that puts extra pressure on the bladder, causing the person to feel like they have to go all the time. If that's the same issue with your son, then he's probably fed up with his body telling him all the time that he has to go potty when he's not ready or wants to go--so he just has accidents. Miralax works (my daughter was on it for years), but if you're looking for something more natural and that doesn't need a doctor visit, then try buying some CALM. You can find CALM at Whole Foods. It is magnesium powder that you put into water and drink. It comes in different flavors. The magnesium softly works on the elimination system so that you naturally and safely eliminate. You can use it indefinitely; it won't hurt the body. My daughter currently uses this when she hasn't gone for a couple days. I usually give it to her at bedtime, and it usually works the next day (or at least by day two).
At the age of 6 years, it might be time to find a good Childhood Specialist who can meet and talk with your son about why he doesn't want to go and work with him to subtly find ways to get him to go to the bathroom. For him, it might be a very real fear that if he goes to the bathroom, whatever game or toy he was playing with might be taken by another child--and he can't stand that (especially if he has any sensory issues at all, this would be a big, big issue/problem for him). A good Childhood Specialist can help him work through his fears, which should transfer over to improved bathroom hygiene.