It might be that something went wrong installing an update. You might try to restart and see if it forces an update. If it seems like it’s hanging on purging extra files, give it a few minutes, that’s not unusual.
You might have a corrupted file somewhere. You can go to the launcher settings and verify your shared cache.
This will attempt to identify and fix any broken or missing files.
Could also be a problem connecting to the Eve Server due to a network issue. Firewalls, anti-virus software, or sever-side security filtering could be the culprit. You might try to connect through your VPN through a different route to rule out a problem with Eve’s security filtering.
You could also try reinstalling the launcher altogether. If none of this works, file a help ticket through the Eve website.