Well, I'm sure that you have already seen the related api docs, but I will link to the relevant documents here:
http://developer.linkedin.com/documents/profile-fields
I do not think that you can do exactly what is after you, but maybe your problem is that you assume that the field will be sent so that the user can hide.
from documents:
Not all fields are available for all profiles. The available fields depend on the relationship between the user you are making, the name and member, the information that the member has chosen to provide, and their privacy settings. You should not assume that anything other than id is returned for this member. If the participant decided to refuse to share data, requests for data return as private for id and last-name. (This means that the identifier is not a unique field.) This affects data retrieved from all APIs, such as personal connections, recommendations, people searches, network updates, or status updates. The code is protected!
For example, documents refer to the person first-name field like this:
This field may return private for users other than the current user logging in, depending on the privacy settings of the participant
It seems that the api setting is not set, and the user has disabled file sharing.