If you have the form set to public, the user does not have to be logged in or even have a Laserfiche account to fill it out. The downside there is that all users will appear as Public/Anonymous when filling out the form. You'll need to collect the form filler's name or email to identify them, and then use that to determine the manager. Based on the sensitive nature of the form, this may not be secure enough, as one employee could easily type in another employee's email address to see their information. 

The real solution for this would be to get participant licenses for each employee in the organization. Participant licenses are less expensive than full user licenses and allows the users to sign into forms to submit authenticated forms. When dealing with sensitive information like health screens, you really want the user submitting the form to be logged in so you know it's actually them, and not try to mock authentication using a public form.