Is there any sort of historical source for this claim? The British navy provided grog to its sailors, but I've never heard this regarding the Mayflower. I am, however, very familiar with the popular historical fallacy that says the ancient Romans drank watered wine because their water was unsafe. Which is untrue.
Also, if boiling in the process of brewing beer is what made the beer safe to drink, why would they not just boil water before drinking it and not bother with the beer, which was more expensive and time consuming to produce?