I may be wrong about this but, to my mind, it sounds like this boy could have
ADD, or even
ADHD (ADD with hyperactivity). Disinterest combined with lack of focus suggests it. Wanting to do things their "own way" is something often seen with ADD/ADHD. I've had success with some ADD kids, if they have some passion for the game. For me, however, this success happened in private lessons -- not in a semi-private classes.
You might ask the boy's mom if her son had been diagnosed ADD or ADHD. But this is something not all parents are willing to admit about their child. If you know who any of his academic teachers are, you might ask them. They might even have some insight on how to deal with this boy.
It sound like you are running a semi-private class for these two kids, correct? Seems to me that the boy's attitude & behavior could be disruptive for the girl -- or even worse, it could be toxic. You might suggest private lessons for these two rather than semi-private. Or a better solution might be to alternate between semi-private and private lessons. You might find that the boys might be more focused during private lessons. Or you might find it easier to deal with his attitude in 1-on-1 sessions. The girl may also benefit from a private class format. OTOH, putting them together sometimes can have other benefits -- social or having another kid their age to rally with.
Does this sound like the 10 yr old boy:
https://www.additude.com/power-struggles-adhd-kids-teachers/
Some other ideas:
https://www.verywellfamily.com/discipline-strategies-for-kids-with-adhd-1094941
https://www.addvantageuspta.com/default.aspx?act=newsletter.aspx&newsletterid=1063
.