Rated 5 out of 5 stars
Manage Accounts dialog opens and closes automatically in a fraction of second, so you cannot enter your account info.
Version: 2.6.1 on Windows 7 x64
Here is the solution..
Somehow previous account information I have saved got corrupted. So, you have to open Gmail Checker SQLite database file (found here: C:\Users\<user>\AppData\Roaming\Mozilla\Firefox\Profiles\<profile>\gmailchecker.sqlite) with any SQLite editor (I used excellent Firefox extension SQLite Manager), open "account" table and delete your account record from that table.
After that, you will be able to open Manage Accounts dialog and enter your account info again.