Brian,
I'm not sure why that would be. The Like button should appear regardless of whether the user is logged in or not.
Can you tell me or test:
* When you have to login, do you have to be logged in as an administrator of the Facebook Application? If so, and it still doesn't show with normal Facebook users, make sure your app isn't in Sandbox Mode.
* If using the JFBCLike module, make sure it's not set to Registered.
* Run the Autotune Error Checker to ensure that there are no detected conflicts or issues on your site.
If none of that helps, can you post or Private Message me the URL of the website where we can see the issue to test?
Thanks,
Alex