If you pay the employee money directly for medical expenses, that is probably going to be taxable income to them. If you pay providers directly for care provided, that may be tax-exempt.
IRC 104(a)(3) says "Except in the case of amounts attributable to (and not in excess of) deductions allowed under section 213 (relating to medical, etc., expenses) for any prior taxable year, gross income does not include—
(3) amounts received through accident or health insurance (or through an arrangement having the effect of accident or health insurance) for personal injuries or sickness (other than amounts received by an employee, to the extent such amounts (A) are attributable to contributions by the employer which were not includible in the gross income of the employee, or (B) are paid by the employer)".
Your EAP is "informal"; it's essentially an understanding your company and the employees have. It's not a formal plan, nor an explicit insurance contract between the company and providers rendering services to covered employees. So it's a question of how to prove to a suspicious IRS agent what the payments were for.
Caveat: Do not rely on my research or post. There may be case law to add more color/nuance to this topic. If you're looking for an opinion you can rely on--and I get the impression you are--I'd suggest getting a tax lawyer to do proper research. Nobody wants to be the one to say "oops" during an IRS audit.
