I'm not sure it is sooo important (especially for a boy, not like for a girl wearing a dress). But if he does get embarrassed he'll change his mind. Or gets his weenie caught in the zipper...
I was a physically sensitive child (and am as an adult) and the seams of socks hurt my feet (like having a splinter), itchy tags made me nuts, I couldn't tolerate certain fabrics. I learned to ignore some of it as I got older, but I still have days when nothing is comfortable.
Your son isn't making it up or being overly dramatic or trying to be difficult. When he says it hurts, he means it.
I recommend reading Raising Your Spirited Child. After I read it, I called my mom and asked her how she dealt with me--she said I had to learn to dress myself very early. Some days I would wear the same clothes because they were the only thing comfortable.
Perhaps you could ask your son to help you solve the problem. Tell him that his father thinks that he needs to wear underwear and you worry about him being embarrassed. Ask him what he thinks might be a solution.
Another brand might be more comfy. Or perhaps boxers?