No you can't is the simple answer.
If you can prove that you have been living together for a significant period then your partner and her son could get Extended Visitor Visas valid 12 months at a time, which would allow her to come and go as she pleased, but she couldn't work or study using that, and nor would she be entitled to free healthcare an so on. I'm not sure how that would work for education, but clearly her son wouldn't be eligible for the public system.
But if you haven't been living together for an extended period then she can't get anything here.
And no you can't employ her as a domestic helper with an EU passport.
Marry her is the easy solution - quick cheap and easy in Hong Kong.