can you try this instead
$strDate = date("Y-m-d");
you can grab today's date and it is of the MySql date format. It work fine with me.
if it is still does not work, try altering your table definition from this
$Query = "CREATE TABLE News (id INT UNSIGNED NOT NULL AUTO_INCREMENT PRIMARY KEY, DateField DATE NOT NULL, NewsField TEXT NOT NULL)";
into this
$Query = "CREATE TABLE News (id INT UNSIGNED NOT NULL AUTO_INCREMENT, DateField DATE NOT NULL, NewsField TEXT NOT NULL, PRIMARY KEY(id))";
hope it will work