Here is an other theory to explain the phenomenon.
I actually don't know if there is any evidence to back it up as I didn't found much on this "leisure sickness".
The idea actually came to me by reading this article on immune system http://www.slate.com/articles/health_and_science/medical_examiner/2014/12/boost_your_immunity_cold_and_flu_treatments_suppress_innate_immune_system.html
The article does not address leisure sickness, but the misguided idea that a boost to our immune system is a good idea. You can read the article for the details, but in short the point is that what people complain about is exactly the immune system reaction. You are not getting the symptoms because of the virus, but rather because your immune system is trying to "cook" it away, while it waits for the body to figure out the proper antibodies to deal with the intruder.
This does not mean innate immune system is bad, but "boosting" it is not so good either. So, my two cents about leisure sickness: since chronic stress reduces the reaction of the immune system, the body gets along just fine without too much of an issue. When relieving the stress, during vacation, the immune system is boosted, which imply it gets overzealous and you begin noticing the symptoms (which are, indeed, the immune reaction itself).
Of course this does not exclude other explanations, such as a psychological consequence of Negativity Bias, which would simply make us notice more negative events (As Kyudos pointed out: but you'd rarely qualify it as "That was a great holiday, I didn't get sick once!")
Again, this is my own idea which sounds reasonable to me, but I don't possess any evidence about this. Maybe it's total bs, but that's my best shot at this time. Hope someone here knows better.
Of course, it is also possible that, in fact, there is no such a thing as "leisure sickness" at all and we are merely dealing with a biased statistical thinking. It wouldn't be a surprise!