Two obvious problems.
It automatically logs you into the websites when you're nearby. When you walk away, it automatically logs you out.
But here's the thing, the Everykey device itself doesn't store your passwords. They're safely stored in an encrypted format on an Everykey secure server.
You should never be able to log onto a secure site without being consciously aware that you are doing so, and you should never have to trust someone else to secure your passwords. You should definitely have strongly encrypted impossible to remember passwords under your personal control at all times, but you can do that better with an ordinary thumb drive and a portable version of a password encryptor like KeePass with a strong master passphrase.
This would not have helped Hillary much if her email was being sent in the clear over the internet to a server that was connected to the internet and open to attack. If all of the email transmitted was encrypted source to secure account on a secure server, then a strong password to log into that system might help in the course of the message, but if the server is compromised then reading email might be possible regardless.