I didn't see the article in question but I wouldn't say it was necessarily incorrect, I certainly don't see how leaving the EU is going to make the illegal immigration issue worse, in fact I'm struggling to see any valid reason for remaining in the EU.
We subscribe to many international treaties and conventions regarding refugees, and would still be beholden to the European Court of Human Rights if we were to leave the EU anyway. Leaving the EU will have absolutely no effect on the ability of the UK to stop asylum seekers (and if we did, it would be internationally condemned - fat chance we'll ever do that), and the chances of us leaving the ECHR is also nil because it's something the British government had a large hand in setting up, and would - again - be internationally condemned.
As for valid reasons, there's the economic one (we are not just paying £8bn a year in government funding for nothing: we get much higher returns in trade - as well as being used as a stepping-stone for English-speaking multinationals into the EU: not something to be sniffed at), there's the matter of keeping European peace, international cooperation and coherent planning strategies, ensuring trade treaties around the world - the EU spends a large amount of its resources on establishing these, and the UK would have to start again all on its own if we left - along with the fact that we would still be beholden to many EU regulations if we left anyway, just as Norway, Switzerland and Iceland are. We'd likely still be paying into the pot if we wanted any sort of special arrangement too (just as those countries do). The idea that leaving is some silver bullet to British freedom (or whatever) doesn't make any sense, and it seems to be something that a lot have pinned their hopes on, as though Britain will suddenly become hugely protectionist and will prevent all sorts of British jobs being lost to abroad etc. Even though unemployment is actually at a 10-year low, and is still very good even if we discount zero-hour contracts.
As for security, the fact is that apart from comments from someone who hasn't worked in MI6 for 10 years (which were also roundly condemned by those who are actually involved in pan-European security today), the EU helps enormously in international counter-terrorism operations. The UK will find it much more difficult to participate in those if we leave, which is something that neither we nor the rest of the EU wants. Fantasies about shutting off borders to stop terrorists makes no sense either: travel for short-stay, which is all you need, will be no easier or harder if we leave.
I've heard the response to many of these points being along the lines of these things either not being necessary (which is false - all of the things listed are very important) or not being something we'd lose if we left because it would be so ridiculous for the UK to stop cooperating on that front (for example: security). But the issue is that we can't pick and choose what parts of the EU we want to cooperate in without paying into it, and being subject to many of the regulations that come out of it. That's the nature of international cooperation which is, whatever your opinion, not going away. Taking for granted the good and saying we'll obviously keep it is extremely naive.
Finally - as for the democratic issue: the European Commission literally consists of people nominated by the member states' governments: i.e. a significant portion of it consists of people our own government put there. There have been almost no situations in which laws are proposed that the UK actually opposes: instead, we ensure that they suit us well. The European Parliament is more democratic than our own, by virtue of how MEPs are elected, and the European Courts are a) not directly part of the EU, and b) aren't elected because electing judges is a really bad idea, and we don't do it in the UK either. Plus, the idea that the EU won't continue to influence and have effects on UK politics if we leave is also silly: the US has considerable influence on UK policy, and an economic powerhouse of a neighbour is absolutely certain to do the same. After all - a very large portion of our trade goes straight through them, and that isn't changing anytime soon.