Unexpected loss of name fields

We were trying to expose username, firstname, lastname only to Admins (Field Conditional Show in Profile Edit). Only the fullname (formatname) stays public.  The intent was these fields would be set at registration and could not be changed afterwards  by the user.
1) When username is required, the user can no longer save their profile without a "username missing" error. The username is still set and remains unchanged.
2) When any part of his profile is changed by the user and saved, his firstname and lastname are nullified. Firstname and Lastname are still required and no error is generated on save or subsequent saves (even with empty names)

I'm guessing you're using conditions to do this? By default CB Conditional will reset a fields value to its empty state if the condition fails. To prevent this be sure to turn Reset to No (this parameter is at the bottom below your conditions). I suggest just setting those fields to Read Only if you do not want them to be changed. You can also use CB Privacy to limit field edit permissions to only moderators.

Ah ha. Yes, I had the Reset option set to "Same as Global", and I'm not sure where the global setting is.

Client wants to be able to edit it from the frontend (where she is accustomed to viewing members).

Ideally, we would like the field to be Read Only for users only, not admins.

You can still edit from frontend using CB Privacy and its edit restrictions feature, but CB Conditional will work fine as long as it's only applying the condition to profile edit and Reset is set to No. The globals for CB Conditional are in its plugin settings.