I think John is right means that I think John is correct (for example, John says that the earth is round - I think John is right).
When we say that someone is 'in the right' we mean that they are morally correct. For example, John says we should help others who are less fortunate - someone else says that no, if they are poor, that is their own fault. We think that John "is in the right". I hope that helps, but if you want to know more, let me know - I appreciate the help you have given me with my Japanese!!