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Here's a solution for MySQL and MariaDB using virtual columns that's a bit more elegant. It requires MySQL >= 5.7.6 or MariaDB >= 5.2:

MariaDB [db]> create table foo(bar varchar(255), chk boolean);

MariaDB [db]> describe foo;
+-------+--------------+------+-----+---------+-------+
| Field | Type         | Null | Key | Default | Extra |
+-------+--------------+------+-----+---------+-------+
| bar   | varchar(255) | YES  |     | NULL    |       |
| chk   | tinyint(1)   | YES  |     | NULL    |       |
+-------+--------------+------+-----+---------+-------+
2 rows in set (0.00 sec)

Create a virtual column that is NULL if you do not want to enforce the Unique contrraint:

MariaDB [db]> ALTER table foo ADD checked_bar varchar(255) as (IF(chk, bar, null)) PERSISTENT UNIQUE;

(For MySQL, use STORED instead of PERSISTENT.)

MariaDB [db]> insert into foo(bar, chk) values('a', false);
Query OK, 1 row affected (0.00 sec)

MariaDB [db]> insert into foo(bar, chk) values('a', false);
Query OK, 1 row affected (0.01 sec)

MariaDB [salt_dev]> insert into foo(bar, chk) values('a', false);
Query OK, 1 row affected (0.00 sec)

MariaDB [db]> insert into foo(bar, chk) values('a', true);
Query OK, 1 row affected (0.00 sec)

MariaDB [db]> insert into foo(bar, chk) values('a', true);
ERROR 1062 (23000): Duplicate entry 'a' for key 'checked_bar'

MariaDB [db]> insert into foo(bar, chk) values('b', true);
Query OK, 1 row affected (0.00 sec)

MariaDB [db]> select * from foo;
+------+------+-------------+
| bar  | chk  | checked_bar |
+------+------+-------------+
| a    |    0 | NULL        |
| a    |    0 | NULL        |
| a    |    0 | NULL        |
| a    |    1 | a           |
| b    |    1 | b           |
+------+------+-------------+
    Post Made Community Wiki by MattW.