A few years ago I changed the domain of a customers website from .eu to .nl because it refused to show up in Google. Anecdotal evidence suggests .eu is not a good domain name for your business.
Some of these are fairly small/obscure; some are fairly significant: europe.eu is the site for a lot of what the EU does; politico.eu is a well known media organisation; coolblue.eu is major Dutch retail business, etc.
.eu domains are not uncommon over here.