I have a ME degree, and an ATP and joined Boeing Flight test as a Test Director on 747-8 and 787. My perspective is from Boeing- each company is a bit different on their flight test branches.

There are several areas of Flight test that you can fit into. An EE degree can be valuable in systems, avionics, instrumentation and telemetry. The other option is to join one of the staff engineering groups and work your way into a flight position. This is usually done by being an expert in an area or becoming an DER/AR.

It does really help to be a pilot and an aviation enthusiast, however they are not always required. What I've noticed that Boeing is looking for is a strong technical skill set, a willingness to learn, schedule flexibility, team player and be in reasonably good health. Any experience you can get in testing (even lab testing) is a plus. An understanding of testing methodology helps. Intern positions do appear- keep an eye out for those as well.

Good luck-

Mustang304