SQLDeveloper
In order to log in to a proxy user in SQL Developer please do the following:
When creating a connection type in your credentials

Then select the "Proxy User" tab
Check the "Use DB Proxy Authentication" box and type in the username of the user you are wanting to proxy into.

We don't need to set the password on this screen, as we are utilizing the username/password on "User Info" to authenticate.
Continue with adding your connection string and save.
Once you log in you can run
Show user;
which will return the user you are proxied in to.
SQLPLUS
In order to access a proxy account via sqlplus do the following
User we have credentials for: "OUR_TEST_USER"
Proxy user we have access to: "AP_CHECKS_PROCESSING"
Oracle DB Instance: "BANPPRD"
sqlplus OUR_TEST_USER[AP_CHECKS_PROCESSING]@BANPPRD
Once you log in you can run
Show user;
which will return the user you are proxied in to.