To become a flight engineer you need to become a licensed AME first. First 2 to 3 years in Aerospace technology depending on the province and the program, then 2 years minimum of apprenticeship. You then must write and pass your Transport exams, when I did them it was 7 exams. After logging enough apprentice time and passing your exams you are then a licensed AME (Aircraft Maintenance Engineer).
If you still want to become a flight engineer you then will need to have a maintenence endorsement on type i.e. 727,747 classic. This is a technical course only, systems, avionics ect.. and is usually between 4 to 6 weeks long depending on the type, this will allow you to sign off that type. After passing this you could then apply for a job as an FE on that type, at which point you would once again go through crew training, procedures and crew endorsement on type ect... However currently in Canada when you are an FE it is considerd a non maintenance job and after 2 years of flying if you have not done any hands on work your AME license will be revoked.
Very few companies still have actual Flight Engineers, one that I can think of in Canada that does is Morningstar (727's), ICC had them (A-300B4's).
Right now most companies in Canada that fly 3 crew aircraft 727's mostly use Second officer's, this is a low time pilot who get's the crew training on the Flight Engineer's station. He will be limited in knowledge as to aircraft system's and such but the salary is about a quarter of what a licensed engineer would make. Engineers are a dying breed, I'm not sure if larger 3 crew aircraft like the DC-10 and 747 classics require and engineer as per TC or not, but we will find out when ACE brings their 10's online soon.There still are FE positions overseas, but they go as quickly as older birds are retired. Hope this helps.