Lump of labour fallacy
flood of cheap labour from Eastern Europe – a boon to bosses, as it suppresses wages
Statements like this demonstrate a clear lack of understating of the labour market in what is known as the "lump of labour fallacy".
Britain had a "booming" low wage sector before EU enlargement but this is almost entirely due to a preference for low regulation and low skilled workers. Immigrants from elsewhere in the EU were attracted by job opportunities, ie. jobs that could not be taken up by UK nationals for various reasons, including the wages offered (there is certainly correlation). If they had not arrived then it's most likely that the positions would have remained unfilled or wages and prices would have to rise significantly. Immigration leads to economic growth through increased aggregate demand (new jobs are created as a result) and indeed many immigrants have gone on to create their own businesses as have so many of the generations before them.
However, while the EU immigrants have been net economic contributors, there is no doubt that they have caused resentment because they compete for scarce and inflexible resources: housing; schooling; healthcare. This was particularly notable in areas with no recent tradition in immigration where, say, even a 5% increased in "foreigners" at the doctor's surgery can have a dramatic effect in waiting times. But they're not responsible for the current problems in the NHS: this is down to the UK's shameful tradition of trying to squeeze a quart out of a pint bottle. Indeed the NHS has for years depended upon skilled immigrants and would probably collapse without them.
Otherwise the analysis is fair (the UK government is still looking for the clue hammer), but largely misses the point. Until the UK formally requests to leave the EU then all of these speeches and remarks are purely for domestic consumption, which is why it contains nonsense like "no deal is better than a bad deal". Game theory alone tells us that a bargaining position of maximum demands is not optimal in the forthcoming negotiations, so we can expect this change. In fact, the speech was reminiscent of May's many speeches about reducing the number of immigrants to the UK. And just how successful was she in this respect? Even immigration from outside the EU, where the UK can do pretty much what it likes, rose.
It's possible this is a long game and the idea is to start making things so unpleasant for the UK in order to sideline the hardliners and make the inevitable compromise more likely to succeed politically: an vote in parliament is likely to depend as much upon the opposition as it is upon the government's own supporters.