My mother moved into a MC facility recently after living with us for 4 years. We developed a friendship with a HH aide who came to our home.

Thank you all for your thoughtful answers. The meeting we were to have today didn't happen. Another meeting ran long and I caught the director on the way to the elevator. I asked if we could reschedule and she said no need, she just wasn't comfortable with her aide helping mom. She avoided my question about whether there was a regulation. I found it very disrespectful that she wouldn't be courteous enough to sit down with me and explain her reasons. But since our friend is her employee we'll have to let it go.

As a former CNA our facility discouraged visiting Residents on days off. Time off was to recharge mind and body. If the HH falls or gets hurt in some way and she is clocked out could complicate who’s responsible. As far as one on one care that was allowed if it was medically necessary, but family could not pick or choose. That was left to the Nurse Manager.
When my mother broke her hip in January 2013 I had to hire outside agency to bring my mother to my daughters wedding in May 2013. Hiring staff for out of facility things was not allowed.

Hmm
At mom's memory care, I've seen families ask staff if they will help out on a one on one basis on their off hours or days

That said, I don't know what the actual policy is as the company would probably prefer one hire their personal attendants but they all work multiple jobs to make ends meet

I would simply ask what the issue is - is she not clocking out before she visits with mom?

Yeah, I'm not sure why they would coin it as "illegal." I think, though, there could be a problem if you or your mother are offering some payment, tip or bonus.

I can see there might be a policy in place to prevent favouritism, but "illegal" is pure hyperbole. And I would question that policy anyway, since I live in a rural area there are limited choices for facilities and it is inevitable that some of the staff have their own loved ones residing there, should they be expected to never visit?
Unfortunately no matter how ludicrous the policy if they choose to dig in their heels things can be made rather unpleasant for your family and the big hearted caregiver, so go into this meeting prepared to be cheerful and pleasant (gag), hopefully you can find some kind of compromise.