Can Americans Teach English in Europe Even Without Citizenship from an EU Country?

Are there summer opportunities to teach English abroad? 

YES, however options will be more limited than if you were looking to teach English abroad for a year or even for six months.  Except for some short-term contract work, most language schools look to hire full-time teachers for 10-12 month contracts.  This means that most short-term opportunities for English teachers will be either in summer camps or in a volunteer capacity.  

Teaching English abroad in the summer as a volunteer or at a summer camp may not pay much (if anything at all), but you can find great opportunities to spend a couple of months abroad and any experience you gain will be a great addition to your resume, especially if you decide to teach English abroad again at a later date.