Try removing the < and > symbols:
CREATE UNIQUE INDEX ix_IndexName ON User(User_Name) WITH IGNORE_DUP_KEY = ON
Or better:
CREATE UNIQUE INDEX ix_IndexName ON User(User_Name) WITH IGNORE_DUP_KEY = OFF
If you want to disallow duplicates.
But personally, I'd do that as part of my user interface code: when a user tries to register I check to see if the username is taken. If it is, tell them. If it isn't, tell them it's fine.
Waiting until the last minute when it's inserted isn't the "nicest" way for a user.