Checkbox fields have a value of 1 (yes) or 0 (no) or NULL (no). It's best to condition Equal To or Not Equal To 1 for accuracy. So to condition against your checkbox field in your auto action you'd have something like the below.

Checked
[FIELD_NAME] Equal To 1

Not Checked
[FIELD_NAME] Not Equal To 1

As far as pushing data to a 3rd party extension we can't help you there. If it's in the database you'd use a Query action and have to write the necessary SQL to do that. If it has an API you can use you'd use a Code action so you can write the necessary PHP to do that.

You don't use PHP tags in a Code action. Just put your PHP code in it only. You also don't use MySQL functions directly and need to use Joomla or CB API to execute a query. You also don't need a Code action for that and can just directly put your SQL in a Query action (it's not jQuery) it self. I can not help you write database queries for 3rd party extensions. I've no clue what their database structure is or what data it expects. Using your supplied queries for example you'd have the below Query action.

Action
Triggers: onAfterUserUpdate, onAfterUpdateUser (this covers frontend and backend profile edit)
Type: Query
User: Automatic
Access: Everybody
Conditions
1: [FIELD_NAME] Equal To 1
Action
Query:
Recommend the below resources to get started learning SQL.

Documentation
dev.mysql.com/doc/refman/5.7/en/entering-queries.html
dev.mysql.com/doc/refman/5.7/en/examples.html
dev.mysql.com/doc/refman/5.7/en/functions.html

Free Lessons
www.codecademy.com/learn/learn-sql
www.khanacademy.org/computing/computer-programming/sql
sqlbolt.com/
www.udemy.com/topic/sql/free/