The E.U. verses Non E.U. Question

First things first, the E.U. question. Yes, it's true, you must hold an E.U. passport to qualifiy for any job position in the territory. Unlike its neighbors, Spain & Italy, the French mostly follow this rule.

However, it is still possible work in France if you are non E.U. if have can obtain the "Carte Sejour". As a non E.U. person, the easiest option is to enroll in a French language school in France. Once enrolled, the language school will issue you an "attestation" stating that you have committed to lessons with them and you can use it to get a student visa. The student visa allows you to work 15 hours a week. At the end of your visa, for however long it is, if you have a job at that moment you can renew your visa as a carte sejour. With this you can legally work full-time. This is probably the easiest route no matter what your age.

If you have mastered the French language and/or have absolutely no interest in long term lessons of any kind, the other option is to become a "Profession Liberale". It means a free-lance professional. While you can do this in any domain, "English Teacher" is probably the easiest to break into as you are a native English speaker, so it will tempt potential clients to choose you over a French person. To do this you apply for a long term visa, similar to the student visa process but longer, stating that you will support yourself as a Profession Liberale. Once in France you can register yourself with URSSAF the government entity that oversees profession liberales. The registration is actually the easiest part of this entire process. Then you are free to start looking for work. BUT, looking for work means looking directly for clients. You're not going to be able to work for a language school unless it's a short term contract deal. You are working for yourself so it would demand a high level of French. So again, if you're not there yet on the French, opt for the student visa choice:)

All in all, the E.U. issue does'nt have to be an obstacle to living and working in France. However, it does mean that more work and money will be required just to set up. Also know that there is no way to avoid the visa paperwork in your home country before coming to France and without a doubt, it will continue when you arrive.