How do you handle pupils who feel their answer(s) are correct and won't change?

Discuss why they think they are right. Start asking or doing the "what if" questions and make sure the questions make the student really think. Encourage student to look at both perspectives (both sides) and then discuss why you, the teacher, think the student is wrong and shed light on the situation. If student still thinks they are right then teacher should try to provide evidence, ask the class, and discuss the situation with one another. You may also introduce another teacher in the discussion and let student hear from adults point of view. Always remember to discuss in rational tone of voice to encourage student to critically think and not overwhelm the student or yourself, the teacher.

Give them the evidence of the book

Keep calm. Getting angry and escalating an argument will not benefit either one of you. You might even end up saying something you'll seriously regret.

Consider the subject you are discussing. Is it even worth arguing about? If you're debating something like the best kinds of pizza toppings, it is probably best to just let it go.

Consider your relationship to the person. Different relationships will often dictate how the situation should be handled.

Use productive language. Just as resorting to anger won't improve the situation, resorting to petty language won't help things either. Countering an argument with something like "That's just stupid," won't get you anywhere and will most likely put the other person on the defensive.

Redirect arguments. If you find yourself constantly having to converse with a person who always has to be right, become a master of misdirection. Change the subject to disarm them, or guide the discussion into more neutral territory. Try to find a topic that you agree on, then you can both be right.

Provide them with proof! but do so in a non-patronising or belittling way so that their dignity is preserved. Appeal to their better nature whilst explaining why they are wrong and where possible give them a little "out" something like: " I can see why you would think that way, it can be confusing sometimes", or anything along those lines, let them come out of it without "losing face"!.

Try to negate the answer by building conclusions.

Immediate correction of students causes fossilization,give'em chance to explore it themselves in course of time

I ask them to approach the matter with an open mind, then research the question and possible answers (via internet, books, etc.), then come back and convince me that their answer is correct. Depending on their age, they might also benefit from writing a short paper on the research for extra credit (even if their answer is proven incorrect in the end). 

In psychology. they should be treated according to their minds and convince them Bmntgahm

I haven't expereinced such pupils actually, but I would get them contented by evidences from the book