I would like to add a BOOLEAN column to a MySQL table which will be named is_default. In this column, only one record can have is_default
You can't have such a constraint in MySQL.
However if instead of TRUE and FALSE you use the values TRUE and NULL then it will work because a UNIQUE column can have multiple NULL values. Note that this doesn't apply to all databases, but it will work in MySQL.
CREATE TABLE table1(b BOOLEAN UNIQUE);
INSERT INTO table1 (b) VALUES (TRUE); // Succeeds
INSERT INTO table1 (b) VALUES (TRUE); // Fails: duplicate entry '1' for key 'b'
INSERT INTO table1 (b) VALUES (FALSE); // Succeeds
INSERT INTO table1 (b) VALUES (FALSE); // Fails: duplicate entry '0' for key 'b'
INSERT INTO table1 (b) VALUES (NULL); // Succeeds
INSERT INTO table1 (b) VALUES (NULL); // Succeeds!