You should also verify that your anti-virus product has been tested against Y2K bugs. The first of January is no different. There will almost certainly be viruses written deliberately to trigger on New Year's Day, but their threat is no greater than that of any of the other tens of thousands of viruses in existence. In fact, perhaps it is lower because of the small number of people who will be at work on that day. It is also important to remember that viruses do not just cause problems when their payload triggers.
Viruses also cause problems when they spread.
Also, if a virus were to infect a company's customers, they may not feel blas about the infection. However, that does not mean they are necessarily any greater threat than viruses which trigger on any other date of the year. This is unlikely, in my opinion. Remember that viruses typically take some time to spread. As most computer users will not be at work on New Year's Day, an email-aware virus like Melissa is unlikely to spread far.