Okay, found these steps in the archived forums:
1) Arrange your characters on each server in the exact order you want them. Make sure to move each character at least once.
2) Go into your CoH folder, find the subfolder with your account name on it.
3) Find playerslot.txt. (This may be tricky, what with the folder having one file for every player you've ever exchanged /tells with.)
4) Right-click, "proprieties", check "Read Only", "Apply".
FYI: Make sure the folder itself isn't set to read-only? That'd override any individual file settings. Thanks to @THEDarkTyger for imparting this suggestion, as it resolved the OP's issue.
In the same thread Impish Kat also suggested the following:
"I just went into the playerslot.txt file, deleted the content, saved the (empty) text file, and let the game rewrite it when I logged in and arranged my characters as desired. Seems to do the trick without changing the file to read only."