Couple of potential issues:
You'd have to get a Mobi Pro
The standard Mobi cards only do .jpg. The Pro adds transfer of video, incl .avi files.
Mobi:
https://mobihelp.eyefi.com/hc/en-us/articles/201153460-Supported-File-Formats
Mobi Pro:
https://mobihelp.eyefi.com/hc/en-us/articles/204131934-Mobi-Pro-Features-and-Technical-Specifications
Card capacity.
When the card is full, that's it. You're done. The earlier renditions of the Eye-Fi cards had a feature (Endless Memory) that would delete files that had already been transferred, at a percentage that you set.
For instance, if the card was 80% full, delete earlier ones.
That function has been removed. So once the card is full, it is full, and you need to erase it manually.
File size.
The videos must be under 2GB each.
You could use IFTTT to opload to Facebook, Twitter, etc. But that evidently requires use of the Eye-Fi cloud, which now requires a subscription
https://mobihelp.eyefi.com/hc/en-us/sections/200835024-IFTTT-Recipes-for-Eyefi-Cloud
However...since you can designate whatever folder you want in the Eye-Fi application, I see no reason why you can't have the DropBox application look at that folder and do the upload by itself.
Camera compatibility
Will it work with that Brinno? Completely unknown. You'd have to just try it.
I looked around, and could not find their database of compatible cameras. Maybe ask them.