I don't think he said that no Mexican (or company based in Mexico) was paying him. He said he would be being paid in his home country. It's extremely easy to pay someone in France from a French account for work performed in Bahrain. It's probably legal, too, as long as all the paperwork is in order. In the case of "no paperwork", I really took this to mean he was going to be paid under the table.
Sure they do. Every civilized country does. They may not like to, but hospitals are going to treat you if you need it and sort out payment issues later. In some cases that may mean they never get paid. The biggest difference is that in most countries access to healthcare is considered a basic human right so people don't get quite as wound up at the idea of a sick person getting "free care". They also tend to have reciprocal arrangements between governments to account for expensive care of a foreigner. Nobody can make such arrangements with the US in the absence of an actual US healthcare system.
In terms of "other benefits", I'm not a Mexican citizen but I am entitled to certain social benefits if I were to need them. I don't need them, so I don't access them, but they're available.
I have no clue what this means. To what "reverse" are you referring? If you're implying that a foreigner can roll up and just get free routine medical care in the US, you are completely mistaken. They're absolutely going to get billed. If they get a prescription they're probably screwed because there's a good chance it'll be too expensive to fill. They may not pay the bill, but in the US basically nobody is entitled to free care. The notion that foreigners are somehow able to access free healthcare that isn't available to citizens or legal residents is complete fiction.
This just does not seem like a vague notion to me at all. You don't just get to show up in the US from somewhere else and expect to be allowed to take a job, even a part-time crappy job, without permission to do so. The people who most presume that happens all the time are among those get the most wound up about it, but those are the very people who should assume other countries are no happier about it. For the hippies who think borders are dumb anyway, they're just going to have to accept that for a range of legitimate reasons (and, sure, some bogus ones), countries restrict who can do certain things within their borders. For those who care about illegal immigration or inappropriate use of a country's resources by foreigners, it should be very easy to understand that "illegal is illegal" everywhere, not just in the US. For those who aren't at all bothered by illegal immigration, it's still the case that many people don't like it at all and that people with such views are prevalent in every country.