Well... what you can say grammatically and what is natural to a situation are not the same.
"I do not wish I served like that guy"
You are watching a tennis match with your friend. You say "You know, I wish I played a better game of tennis." The "guy" serves into the net four times in a row. Then you say, "But I don't wish I served like that guy." -- This specific situation is okay.
In most cases "I'm glad I don't serve like that guy" (you know you don't) or "I hope I don't serve like that guy" (you're not sure how good your serve is) or "I wish I didn't serve like that guy" (because, in fact, you do) would be more likely.
"I wish I could not fly" would only make sense if you can, in fact fly, and wish you couldn't. It seems an odd wish. Maybe you really don't want to go to your brother-in-law's birthday party in Las Vegas and would love to say "I can't get in there in time. Sorry!" but you know you could, because you could fly there if you really wanted to go. What do you think this means?
"I wish I wouldn't" is okay too. Let's say you're really shy. "I wish I wouldn't get so nervous when I have to speak in public."