There's some research to support that taking vacations is beneficial - see http://www.tandfonline.com/doi... (paywalled, sorry) and http://joh.sanei.or.jp/pdf/E51... (pdf alert) - but I get the feeling that the debate is not about whether vacations are good or not but rather about if it' should be the government that gets to dictate that vacation time should be granted.
In the following, I will make some assumptions:
1) Taking vacations reduces stress and provides health benefits (sort of based on the research cited above).
2) If the reduced stress and the health gains are significant enough, the government will either have to spend less on health care or social benefits, or the government will earn more in the form of taxes.
3) The employers are willing to exploit lower paid workers for short-term financial benefit.
4) The government is willing to act in the interest of the citizens.
Granted, the second assumption is the weakest one but at least it sounds logical to me. The time that a person spends being sick or recovering from a sickness is time the person will not (or cannot) spend working / earning money. Less money earned should directly correlate to less taxes paid which of course means less money for the government. Furthermore, less time being sick probably means less money spent on medical assistance (be it financed by the person, an insurance company or by the government).
The fact that employers are willing to exploit workers is in my opinion quite clear. There are quite a few regulations put in place by the government to minimize that - for example regulations requiring protective clothing when handling hazardous materials. A better example is probably the minimum salary, which (at least in theory) should provide all workers well enough remuneration to survive on it. If there's a guy willing to work 60 hours per week for $200 per week flipping burgers and an employer willing to hire that person, should it be in discretion of the employer and employee only to decide if that is acceptable?
Based on the previous points it is my opinion that the government should get to mandate a number of paid vacation days because it provides substantial enough benefits for the citizens. Since the employers would not otherwise provide the paid vacation, a law stating such a requirement should considerably increase the number of employers offering paid vacations, assuming that penalties of not doing so were sufficiently high.