Yes, you can. For the most part, you will need a lot of PIC time in the aircraft they are looking for. There are some overseas CFI jobs that are easy to get, but the QOL will be bad. Places like China, Saudi Arabi, etc...
With an FAA license you can only fly N registered aircraft, and the same holds true for Europe. If you have a JAA certificate from one EU country (and Switzerland), you can (generally) fly in any other. So if you train in the US, you have to convert your license to a European country's license.
Now most European carriers require a "Frozen ATPL" (which is like having taken the American ATP written but not having 1500 hours), except this requires taking 14 written exams and sitting in ground school for 700 hours.
There are other options but most are unrealistic for low time guys. Only exception I know of is that Ryanair (when times were good) hires foreigners and the Irish gov't validates FAA ATPs. Make sure you have the right to work in Europe too.