Name changes happen at the beginning of every season (that means at the beginning of March, June, December and September), so they happen four times a year. However, Pokemaster was extremely late with the last December one which went up in late January, so he made the next name change period June. You'll have to wait until then.
And no, you do not have to change your name. This is only for people who want to, you aren't forced to change it if you don't want to. Pokemaster does the name changes by opening the "username" field in the My Account page for you to edit, so yes, you can change your name any amount of times for the duration the name change period lasts (usually a few days). Just don't abuse it, and remember that you're stuck with the name you're using when Pokemaster removes the ability to edit usernames.