If you ruled out medical problems and discipline hasn't been effective, there is a good chance that your son has an emotional problem. I would think it best to get your son evaluated by a mental health professional before this problem worsens. You could also discuss this situation with the school counselor.
Very, very good points by Annie!
Also, there are some good online resources on encopresis. A lot of times if a kid has it, he can't even feel the liquid poop slipping out, around the blockages. I don't think he is doing this because he is lazy, either. Or to be defiant. Or even doing it on purpose. Put him in pullups and say cheerfully that you think he has a medical condition and he can wear these until you get him in to see the doctor. Act confident, it will probably help him a lot not to be feeling like a bad kid about this. In other words, odds are very high that this is a medical issue, so take the shame load off!
If your pediatrician seems useless or stumped, ask for (I would demand) a referral to a pediatric gastroenterologist. Do keep putting you son on the loo in the meantime as you are doing -- 10 minutes tops, each time. Not all day. All that will do is waste everyone's time and give the poor kid stiff lega and a frozen butt. I would ask the pediatric GI doc to do another x-ray specifically with encopresis in mind, and also ask him to explain the working of the colon -- once a person has fecal impaction, the colon just can't contract properly. If your son has this, it's out of his control.
If this is the issue, he might need to have impacted stool removed at the hospital, and AFTER that, it is very important to give him a stool softener every single day. Not a good idea to start such a regime if he does have a lot of impacted stool, since it will create a lot of pressure behind the impaction. But once you know his lower intestine is cleaned out, it pretty much feels lifesaving for a kid and family, if his stools come out at the normal pace and texture every day. They make daily gummies (fiber gummies, not vitamins) that work well to keep the kid regular once everything is cleaned out. You have to stick with fiber supplements for a year at least, after you're sure there is no impaction, just to get the intestine to heal and be able to contract normally again.
Again, be sure you know the state of his lower intestine, bowel, colon -- before you do any stool softening, and the way to do that is to get the poor kid back to a decent doctor. A lot of pediatricians really don't understand encopresis, and it is a common enough issue that they should.
Obviously the punishing and rewarding haven't worked very well. So knock off the punishing as that certainly won't work.
I do know from my experience as an elementary school principal and teacher that some kids just won't use the bathrooms to poo. The older kids in there scare them. The teacher needs to let him go during class time. Having him sit on the toilet after breakfast and dinner is a good idea. But if his poo schedule happens around noon time - it won't help at all.
How long ago did you have the xrays done etc? Because if it has been awhile, there could be something going on now.
The link I have posted is on fecal retention. It pretty well covers the whole problem. It is definitely worth your time to read as you need to decide if it is something like fecal retention or just not being able to use the bathrooms. The link is - http://www.practicalgastro.com/pdf/November02/LoeningBauckeArticle.pdf
I do have a hard time believing he is doing this just because he is lazy. I would think that there is something else going on. One of the keys is (as the link explains) what his poos are like. Knowing that really helps determine the problem.
Hope this helps.
try putting him back in pull ups? and then let him look buddy your not a baby any more.. any babies wear diapers. see what his reaction is.. if it is him just being lazy then get him to sit on the toilet most of day untill he poops. once he does reward him. Then tell him this is where he has to go or else hes guna be sitting in poop all day and being picked on at school. maybe get his dad to help him make him proud hes pooping like daddy does. if youve already tried all this then idk hun. but then best thing to do is to have him in pull ups at least at school. Hope things get better for him and you :)
keep us posted.