Solution
NI no longer has SHA-1 certificates for driver signing and now only uses SHA-256 certificates. Windows 8.x and 10 have built-in support for recognizing drivers signed with SHA-256, but with Windows 7 that functionality was introduced with a hotfix, KB3033929. Windows 7 (and Server 2008 R2, and WES7) systems that do not have KB3033929 installed will not load NI Drivers. Some NI drivers are necessary for system boot, which means the system will fail to start without the Microsoft KB installed. A system with all Windows Updates should not encounter this issue.
To resolve this, press F8 at boot and start the system with “Driver Signature Enforcement Disabled." After which, install the Microsoft KB3033929: