Did you see the note that "Report an Issue" is for reporting possible software bugs? This post would have been better served if it had been posted to the Filemaker Pro section. (See file tab at top of your screen.)
First a point of clarification: FileMaker doesn't ask for a password when you launch the application. It asks for a password when you attempt to open a file--in this case a remote access to a hosted database file--that has passwords defined via Manage | Security. That is what you have going on here, correct?
These are pretty obvious things, but should be ruled out:
Can you confirm that the problem Mac is opening the same copy of the file as the other macs?
Any chance that the user on the problem Mac is using an "opener file" that opens itself with a password set in File Options, then opens the remote file before closing itself? (If so, you might not have noticed this file as it wasn't open to check.)
Are you using FileMaker defined passwords or server based security?