Currently, the Issue Collector doesn't allow use of a number of fields in JIRA. It seems mainly to be since they are not allowed to be accessed by anonymous requests, per Using the Issue Collector:

• Not all fields of types of fields can be added to the form, since some fields cannot be displayed to anonymous users. The fields types that can be displayed are:
  • Standard Fields: Summary, Description, Components, AffectsVersion, Environment, Priority, Attachment
  • Custom Field Types: Date Time, Radio Buttons, Multi-Checkbox, Multi-Select, Number, Select List, URL field, Version Picker, Cascading Select, Project Picker, Single Version Picker, Text Field, Free Text Field

This issue was created to investigate/consider the possibility of extending the Issue Collector to make use of functionality which is only available to authenticated users.