The bill is much less far-reaching than suggested. Here's the most relevant part of the actual text:
"21-W:4 Mandatory Use of Proprietary Software Prohibited. No person in the state of New Hampshire shall be required to use proprietary software for any interaction with the government, including but not limited to: the filing or payment of taxes, remote appearance for court proceedings, the taking of standardized tests or the completion of coursework by school students, applying for or receiving unemployment benefits, or other similar benefits, unless the government agency has determined that the proprietary software is the only means available for the required interaction. In such cases, the agency shall post a notice of its determination and the use of proprietary software on the agency's website.
21-W:5 Proprietary Software on State Websites. No person in the state of New Hampshire shall be required to allow the execution of proprietary javascript in their browser when visiting websites administered by the state of New Hampshire. The department of information technology shall verify that this is the case for each executive branch agency via the use of browser extensions of their choice that are designed to detect and block the execution of proprietary javascript in their browsers, and by confirming that no proprietary java script blocks the functioning of any third-party online archiving services that are attempting to archive state websites."
The ONLY cases where it's banning non-free software is where the non-free software touches the citizens directly...where those citizens have to use or buy it. Could be tricky for some types of interactions, like allowing state employees to get paycheck stubs/records, Zoom-based court procedings, etc. But it's not broadly "banning non-free software".
https://gencourt.state.nh.us/bill_status/legacy/bs2016/billT...