Self-governance is one of the core principles of classical republicanism. Free and Open Source Software (FOSS) is closely related to the idea of self-governance and hence classical republicans should promote its use. The opposite of FOSS is proprietary software.
Why is proprietary software a problem for classical republicanism? Since the source code of proprietary software is usually closed, users of such software are dependent upon its supplier for fixing bugs and other useful updates. This dependency decreases the ability of communities to exercise self-governance and increases the power, and consequentially the potential of power abuse, of software companies.
On the other hand, free and open source software enables members of the public to contribute to the development and improvement of software. If a supplier of a certain software package would decide to quit its development, its role could easily be taken over by other developers. In the case of proprietary software, however, this would be illegal as it would be an infringement of intellectual property, unless explicit permission would be granted.
Given the ongoing process of digitalization of society, the need of FOSS is greater than ever. Modern governments are virtually impossible to run without computer systems and associated software. By using free and open source software, the state is able to prevent becoming a hostage of software developers, which could under mine public interests.
Also the use of proprietary software is a (potential) national security risk, as software developers could hide back-doors in essential software. Back-doors could be abused by foreign powers and terrorists to disrupt society. FOSS allows people to inspect code on possible back-doors and the subsequent removal of those upon discovery.