We have no "right" to happiness according to C.S. Lewis. Does this mean that we shouldn't be happy at all? Does this mean that we shouldn't seek out things that make us happy? Yes and No.
C.S. Lewis starts out with an example of a man who divorces his wife in order to marry a woman who divorced her husband so she could marry the man. Clare, his opponent in this particular argument, claims that he has every right to do so. Everyone has the right to do what makes them happy; everyone has the "right" to be happy. Before Lewis attacks this absurd statement, he attacks his opponent by saying that he knows for a fact that "all of her friends hate her." (One could say that he was being "Bulveristic" at this point).
Lewis goes on to say that a "right" to happiness is as silly as a "right" to be a millionaire or a "right" to be six feet tall. After all, if being a millionaire or six feet tall are the only things that are going to make you happy, then you would have a right to be both of those things according to Clare. Since being six feet tall is impossible without the right genetics and only a few people become millionaires, it would seem that your "right" to happiness has either been taken away in a few instances or it does not even exist in the first place.
Next, Lewis talks about law: Laws of the government and moral law. Both he and Clare believe in both of these. It's just that Clare seems to think that a right to happiness fits in the moral law, and Lewis disagrees. Lewis would probably agree with Clare that people should pursue happiness, however. He would also say that if you do not want to be happy then there is something wrong with you. It's just the manner in which people pursue happiness and what bounds should be placed on this pursuit that are disagreed upon. Clare would say that there should be absolutely no bounds place on the pursuit; it is, however, a "right". Lewis would say that pursuing happiness is alright as long as you are within legal and moral laws (God's laws). After all, who can bring true joy but God. Only by walking with God and doing what He wants can you find life long joy and true contentment.
Lewis proves his position on the pursuit of happiness with two consequences of the belief of a "right to happiness". First, women in our society will surely loseout in this situation. Because they tend to be more monogamous in nature, they will be left time and time again by men who are pursuing their "right to happiness". Also, since at the time Lewis was writing this, women in the workplace was far less common, they would also lose out economically if their husbands left them.
The next consequence is that our civilization will be swept away. Everyone will be doing what they want and have no regard for what is best for the communities they live in.