The employee needs to not burn bridges. They need to walk on eggshells through the exit interview. They need to maintain this secrecy long after leaving!
There's a lot that I would love to tell my past managers. Stuff I might tell a trusted co-worker over drinks, or would only tell non-work friends and family, because I know people who have been burned by opening their mouth and being too honest.
If you're a company owner or a manager with a lot of sway in your company and want to fix this, here are a couple tips:
1. Be transparent and honest: criticize yourself and your company when you deserve it, in emails or memos, not just verbally. A culture that tolerates introspection and criticism is more likely to tolerate external criticism.
2. Do not be completely honest and transparent when discussing occasional failures of past employees: it's all too easy to remember mistakes and forget the thousand times they got it right. Make sure that your current employees will know they will be spoken of positively after they leave.
3. Make sure that your employees are friends with each other (and yourself, if possible) outside of work. Everyone needs friends, it's hard to make friends as an adult, and even Gallup uses "do you have friends at work" as an indicator of company health. This will mean that employees will have social conversations during and around work: this is good, they will be more productive if work is a happy place and not sterile, and the cost of a few minutes here and there is much less than having unhappy, lonely people trudging through the work day. If they are trustworthy friends they will be able to communicate honestly.
4. If you are told something negative, don't take it out on them, fix the problem!