No.

Only the timestamp and datetime datatypes support the on update clause.

In recent versions of MySQL though, you can combine an auto-updated timestamp column with a computed column that turns it to a unix timestamp - which avoids the need for a trigger:

create table mytable (
    id int primary key,

    -- auto-updated timestamp colum
    update_ts timestamp on update current_timestamp,

    -- unix timestamp computed from the timestamp
    unix_update_ts int as (timestampdiff(second, '1970-01-01', update_ts))
);

Note that we cannot use unix_timestamp() in a computed column as of now; MySQL seems to assume that the results of this function are not deterministic (which is not correct when the function is given a fixed argument) - but we can work around this with timestampdiff().

You can do this with a DATETIME or TIMESTAMP column:

CREATE TABLE mytable (
  id SERIAL PRIMARY KEY,
  timestamp TIMESTAMP DEFAULT CURRENT_TIMESTAMP() ON UPDATE CURRENT_TIMESTAMP()
);

Then you can query it to get the UNIX timestamp like this:

SELECT UNIX_TIMESTAMP(timestamp) AS ts FROM mytable;

MySQL 5.7 has generated columns, but it doesn't work with nondeterministic functions:

mysql> ALTER TABLE mytable ADD COLUMN ts INT UNSIGNED AS (UNIX_TIMESTAMP(timestamp));
ERROR 3102 (HY000): Expression of generated column 'ts' contains a disallowed function.

You could define a trigger:

mysql> ALTER TABLE mytable ADD COLUMN ts INT UNSIGNED;
mysql> CREATE TRIGGER ts BEFORE UPDATE ON mytable FOR EACH ROW SET NEW.ts = UNIX_TIMESTAMP(NEW.timestamp);

mysql> select * from mytable;
+----+---------------------+------+
| id | timestamp           | ts   |
+----+---------------------+------+
|  2 | 2020-07-23 23:09:57 | NULL |
+----+---------------------+------+

mysql> update mytable set id = 3;

mysql> select * from mytable;
+----+---------------------+------------+
| id | timestamp           | ts         |
+----+---------------------+------------+
|  3 | 2020-07-23 23:14:05 | 1595546045 |
+----+---------------------+------------+

You should create a matching trigger before insert as well.