Unfortunately no, MySQL 5 requires constants for the default. The issue was discussed in much more detail in the link below. But the only answer is to allow null and add a table trigger.
MySQL only recently accepted UUID as part of their DB package, and it's not as feature rich as we'd like.
I'm not sure if the above answers are for an older version, but I saw somewhere that you can do this using the unhex() function. I tried it and it works. (maria db version 10.2)
You can do
.... column_name binary(16) not null default unhex(replace(uuid(),'-',''))
and it works. To see the uuid just do hex(column_name).
Note that MySQL's UUID() returns CHAR(36), and storing UUIDs as text (as shown in the other answers) is obviously inefficient. Instead, the column should be BINARY(16), and you can use UUID_TO_BIN() when inserting data and BIN_TO_UUID() when reading it back.
CREATE TABLE app_users
(
app_user_id SMALLINT(6) NOT NULL AUTO_INCREMENT PRIMARY KEY,
api_key BINARY(16)
);
CREATE TRIGGER before_insert_app_users
BEFORE INSERT ON app_users
FOR EACH ROW
IF new.api_key IS NULL
THEN
SET new.api_key = UUID_TO_BIN(UUID());
END IF;
Note that since MySQL doesn't really know this is a UUID, it can be difficult to troubleshoot problems with it stored as binary. This article explains how to create a generated column that will convert the UUID to text as needed without taking up any space or worrying about keeping separate binary and text versions in sync: https://mysqlserverteam.com/storing-uuid-values-in-mysql-tables/
As of mysql v8.0.13 it is possible to use an expression as a default value for a field:
The default value specified in a DEFAULT clause can be a literal constant or an expression. With one exception, enclose expression default values within parentheses to distinguish them from literal constant default values.
Harrison Fisk's answer was great when it was written, but now it's outdated.
Nowadays you can use an expression as a DEFAULT value. This is supported since MySQL 8.0 and MariaDB 10.2. Note that, if you're going to use non-deterministic functions like NOW() or USER(), you should not use binlog_format=statement.