My bigger problem is too short time span of the experiment:

“The intervention period was 30 consecutive days from January 1st-30th 2015. During the following 60 days January 31st-March 31st 2015 participants of all three intervention groups were instructed to shower as preferred.”

Indeed. The whole article has a strong whiff of confirmation bias to it.

And the time of year - I think absence (for any reason) is lower in January anyway as a lot of people have just come back from a week or two off.

We also don't know, but can highly suspect that the subjects in the cold shower group knew or could guess, what was wanted from them in this study. See, Prison Experiment by Zimbardo for the most extreme example, but people will behave in the way they "should" when you tell them the reason of your study.

The fact that the illness days didn't decline makes this study less viable. I'd like to see a followup, but it's hard to imagine a solid base to keep the participants in the dark. It should also be much longer than 30 days.

You couldn't have a true "blind" control group, as there is no way a person could not tell that they are being showered by cold water as opposed to warm water.

Well, you could sedate both groups before the shower and dry them before waking.

I agree to the spurious claim. To shower in the cold is willingly doing something unpleasant. That in itself feels like it is going to affect their attitude towards coming into work -- when doing so is unpleasant.