At present, FreePBX supports the ability to send a call out with a blocked caller ID. If you want to do so, you must format the Caller ID as follows:
"hidden" <2125551212>
(where 2125551212 is the Caller ID you want to send and the word "hidden" causes FreePBX to force the blocking option).
This feature is documented in the tooltips for the Caller ID field on a trunk, but not on similar tooltip for extensions or routes. As a result, I suspect that the feature is not well known.
I recommend that this information be added to the tooltips for the Caller ID field in the extensions/users module and in the outbound routes module, just as it is already in the trunks module.
Even better would be to create a separate field called "Block Caller ID" with a Yes/No pull-down in the trunk, extensions/users, and outbound routes module.