Here are some clarifications (based on further information from the BLS):
As I wrote before, if the government shutdown continues through this coming week, then the unemployment rate in the January report will be negatively impacted. This is a key week since it is the reference week for the BLS report (contains the 12th of the month). If the shutdown continues through next weekend, Federal employees who are on furlough will be counted as unemployed in the January report (CPS, Household survey).
If the government shutdown continues, then the unemployment rate will probably bump up to 4.0% or 4.1% in the January report.
As far as the headline jobs number from the CES (Establishment survey), the jobs were people who are working without pay will still be counted. For the furloughed employees, it is different. Since they are not being paid, the positions will not be counted – UNLESS – legislation is passed that provides for back pay. If the legislation is passed, even after the reference week, the furloughed positions will be counted in the CES (headline jobs number). This is what has happened in previous shutdowns.
So, for the unemployment number, it depends on what happens this week.
For the headline jobs number, it depends on what legislation is eventually passed.