Watch the attitude John.

Everyone here who posts and reads these posts KNOW that it's all just "Opinions". You can take it or leave it. No one here pretends to know what is right for each person or each situation. We just advise by our own experiences and knowledge. No one here is trying to change anyone's world, just make it better for them to live in by empowering them with knowledge!
I do find it unusual for a HOA to have so many employees. However, it's based on the fact that MOST HOA's I know don't employee anyone except sub-contractors. Sub-contractors are Management companies, landscapers, or other amenity caretakers. It is a bit out of the "norm" of what the posters here deal with. It doesn't mean it's wrong, just not familiar with that setup as much.
The HOA is a corporation and the homeowner's are the stockholders in that corporation. If you were working for that corporation, would you not want health insurance benefits provided to you? Knowing it will cost you all money to support but is it not the right thing to do? I think if you provide employment you must also provide the benefits that go along with that.
As far as making sure none of them are "illegals", the group that is submitting the information should be doing that. I highly doubt they are going to provide that personal information openly amongst everyone. That's part of the "personnel files". You can't judge if they find the person illegal or not from the surface. Eventually, the federal laws will be in place to make sure these checks you request are done. Until then, you will have to accept there could be flaws in the system.
I will tell you that not all foreign workers are "illegal", they could be "migrant" workers. That means they come here to the U.S. for a few months and have to return back to their home countries. Migrant workers may be eligilble for some benefits. It may depend on the laws of your state or city.