The Administration would base theirs off a decimal system. They'd divide the year into one hundred, and 'days' into ten, and 'hours' into one hundred. This would be for ease of recording when events occurred. However, as this does not in fact conform to the natural system of days and years, this is not used in normal, everyday life.
For this, I suggest we don't base it so much on religion- not just because there are atheist nations, but also because there are many different unrelated religions. I propose a more arbitrary 'common era' style date- for example, maybe the most powerful nations just decided on a date in order to help trading. This would in my opinion be more likely than a random religion deciding that they'll date everything by the birth of some other religion's deity.
Another suggestion could be that it's based on the fall of some large empire- perhaps a large swathe of the continent was part of such an empire, meaning that when it split into multiple states, it had huge, long-lasting effects on that whole region. Other nations (who hadn't been part of the empire, and so had been relatively unaffected) would have to adopt their dating system, as otherwise they'd just be placing barriers over their trade- sort of like how Chinese people have adopted ACE for international functions.
As for months, if we didn't want to just copy the dating system we have nowadays, we could have thirteen months with twenty-eight days each. There would be a leap year every ten years, I think. However, this could just unnecessarily complicate matters; it's really up to which you prefer: being different enough from reality, or being easy to use.
In terms of naming the months, please don't just think of cool names. It would be great if there was actually a reason behind each name, and amazing if they were tied to our nations (e.g. Machina as a month, named after the church of the machine...)