[Chia.]

Unfortunately I don't know exactly how EAP works.

If you need someone to talk to about the suicidal thoughts then i'd like to suggest the samaritans (free). You can contact them anonymously, you don't have to give any personal details. You can give a fake name.

Although good ethics say that your confidentiality should only be broken if you are at a serious risk , I would like to highlight that this is literally a matter of ethics and there is little, if any, legal protection for you. It all depends on the person you get on the other side of the phone and if the EAP people are apart of the same company then it could all come down to workplace politics and power trips.
As long as the person does't write about you or identify you publicly outside of work then there is little you can do about it if they did decide to tell someone about your suicidal thoughts.

If you really want someone to talk to face to face then the safest option would probably be seeing a private counsellor, although this will probably be expensive. You can research them, make sure they are a member of a professional body, have lots of experience (don't pick someone newly qualified because they will most likely panic and be over cautious at someone talking about suicide) and talk about your need for confidentiality in your first session.
Talk to them clearly about your needs e.g. ask them if they will agree not to write specific things in your notes etc.