Proprietary software should not be used in any sector of any government. It is handing over control of vital technology and information to privately held companies that are interested primarily in profits rather than delivering privacy safe software and hardware.
Governments typically hand a lot of private information that needs to be stored effectively. Without the means to study and modify the code, there is no way to guarantee this information will be done. Malicious backdoors can be installed and exploited for monetary or information gains without the government or the citizens being aware.
Not only can using Free Software ensure the security of it citizens but it can also secure the government itself again other malicious players such as other governments. While the idea of tying up important internal secrets with proprietary software may sound good from a chain of responsibility point of view - having software that can be hardened through Free Software is a much more noble and potentially secure solution.
The services that government provides are funded by its citizens and businesses of society and should they treat them as the highest priority. To use public money to fund private technology that is non-free is essentially handing over control to bad players. It can allow non-free software developers to have a power play over governments. If governments use and fund free software then everyone can gain from this both directly and indirectly.
To do this is to provide an example to others how to live. To grow and share with the community, to lead by example that this is a good thing to strive for.