Today I can't get to my Gmail accounts

Google had an outage today, which also did affect Gmail.

a window pops up

You either have a Gmail IMAP account, or a Google Calendar, or both. In any case, both, IMAP and Calendar use OAuth2 for authentication. When using OAuth2 Thunderbird saves an authentication token underneath 'Saved Passwords', not your actual Google account password. Presumably because of today's outage the OAuth2 authentication token was invalidated, and hence you got the prompt to re-authenticate. You must have done that before, otherwise you'd not have had access to your account before.

Why should I trust Google with all my eggs?

Not sure what you're talking about. You already have a Google account, so that means you'll have to trust them to a certain degree.

The way they ask is aggressive

Feel free to complain to Google. Don't hold your breath though.

Thanks c1, won't expand your name for obvious reasons, that explains what that oauth... thingy was in amongst my passwords.

I had the same problem for a few days now and could not get my T-Bird email to work. Once I understood the problem described by c1, I then knew it wasn't a scam so I went ahead and logged into my Google as per the dialog box and it proceeded to update my email accounts. After all the backlogged messages were downloaded, then T-Bird worked as normal again. Thanks for the explanation and the help!

Old Crooked Cap