A user can be prevented from removing preset criteria - for example "Agent ID" and "Office ID" criteria, by simply hiding those options using CSS on the search widget and results widget.
This will force a user to continue to search based on any applied criteria.
In these cases, it may be appropriate to create a special page just for searching in this way.
Example
To hide the Agent ID and Office ID criteria badges and also hide the "New Search" button, use this CSS:
.bfg-criteria-office-id, .bfg-criteria-agent-id, .results-new-search { display:none !important; }