I configured a hudson instance and have created jobs. While creating builds, I was able to see this option "Trigger the build by accessing this URL + SecretTOKEN" option. Now, I am unable to see that for any new jobs I create. Am I missing some setting or a configuration? The only change I made was running the servlet container from Root to a regular user.
In my Hudson (1.349) when creating a new job, there's an option under Build Triggers called Trigger Builds Remotely (e.g. from scripts), which when selected allows you to enter an Authentication Token (which has the descriptive text from your question).
I'm not aware of any system configuration that disables this. It doesn't sound like a Hudson user permissions problem if you can have permissions to create/edit jobs.
What appears under Build Triggers when you create a new job?