In my case this seemed to be caused by a 32-bit installation of JRE (Java Runtime Environment) I had installed for an unrelated application. With JRE installed it's possible that Flash Builder was attempting but unable to use the 32-bit dlls?
I installed the 64-bit version of JRE alongside it which appears to have resolved the problem.
My previous proposed solution (of installing the 64-bit JRE) resulted in a different problem. After multiple attempts and restarts I was unable to work in the Flash Builder IDE as it became unresponsive on load. Each time I had to use Task Manager to close the application.
Close all applications running in Windows
Uninstall all versions of JRE (Java Runtime Environment)
Uninstall Flash Builder 4.7
Reinstall Flash Builder 4.7