This is not true. In the EU there is a freedom of movement of workers. EU citizens from another member state need to either have an employment contract or they can be told to leave the country after six months of stay.
That is not true. The only thing they need to do is to register their permanent residence in the new country and they can't be forbidden. You're misunderstanding the law, this is done to stop people from not registering their real addresses for tax/avoiding legal action purposes.
EU nationals can even vote in municipal politics in their new place of residence even if they're not citizens of that country; another big argument of the opposing party.
The right to movement of workers comes from the treaties and the ECJ decisions[0]. The basic outline is as follows:
1) You have a right to stay in another member state (host) for three months without any restrictions.
2) After three months, if you are not (self-)employed, restrictions can be placed on you. In particular, hosts will require you to prove sufficient resources to support yourself, including health insurance. If you are not employed and cannot prove sufficient means, you can be removed.
3) If you lose your job, you have a right to remain as a job seeker for six months. If you do not find a job within six months, you have to prove to the host that you are actively seeking a job, and that you have a realistic chance to get one. If you cannot prove that, you can be removed.
4) You get a right to permanent residence in the host after legally staying there for five years.
5) It is currently unclear if you have a right to the six months of job seeking before you've had an employment contract in the host state.
While I'm not a lawyer, I have used these rights myself three times now so I've read through them carefully as well as the local regulations in the countries I worked in.
In theory host countries can require you to prove sufficient resources to support yourself, including health insurance. The problem is that this only really works for countries whose healthcare is structured as insurance in the first place, like the continental European states do; no-one has figured out a sensible way to apply this to the UK's single-payer, free at the point of use healthcare.
Non-EEA nationals who are here long term just pay a surcharge for the right to use the NHS, but that's not allowed by the EU rules. In fact, the European Commission's position is that since the NHS is free, this rule should be interpreted as allowing everyone who comes here from other EU states to use it for free even if they're not working and not paying into the NHS, and that this should fulfil the insurance requirement.
There was a bunch of press controversy over the UK enforcing the rule as written in some cases a year or so back too.
What? UK cannot just ask for an income statement or the you live as a proof, that's what you're saying?
Moreover the migrant worker has to pay tax in the country they're living in, so tax statement may be good enough.
Most of EU has "single payer, free at point of use" healthcare rather than US model of insurance.
And if they're not working, they can be elected after three months. Is three months slip enough to even put a dent in NHS budget? I'd say not at all. It's not like people are migrating to UK to get temporarily better healthcare. (It sucks compared to almost every other EU country.)
Additionally, for temporary stays, UK can probably demand reimbursement from the member country of origin.
No, the UK can't ask for an income statement - the rules only require that people can prove they have enough money to live on. No, they don't have to pay tax - that only happens if they have income. No, this isn't limited to three months - that's only the point where they have to prove they can support themselves in order to remain. And whilst most of the EU doesn't use the US model of insurance, the state-run healthcare in the other EU states is generally set up as a state-operated "insurer" which you have to pay into for coverage. The UK is pretty much unique in funding healthcare from taxes and giving access to all residents for no extra charge.
Which countries practice this?