I have an EAP programme in work and have used it recently.
As far as I am aware all EAP's are actually run by a 3rd party (ie. not VHI, or your company) Although they do ask for details, they confirm that it is only for their records and to allow contact and dealing with your issue.
If i remember correctly, they also state that no information will be provided to employer or anybody else, other than on a generalised statistical level - x people used for housing, y for legal advice, z for mental health etc.
I don't know (and am not asking) what your personal situation is, but I believe these programs deal with a lot of personal and sensitive information on a daily basis, and cannot imagine they would last long if confidentiality was not respected where promised.
I would suggest tracking down the literature provided by VHI on the service and read through it fully. I suppose only use the service if you are convinced it is confidential enough for your needs. Personally I found the service extremely useful and would recommend it to anybody considering it.