I was thinking about this at work (where I seem to do all my thinking about AH), and I really figured out the nuts and bolts of it.
Many people have asked the obvious question: Foreign Legions are created to deal with foreign commitments that an army can't fill on it's own. What commitments would the Canadian military have that it can't fill?
We ask this question thinking in the old French and Spanish model: trying to adjust history to make Canadian colonies, etc. But that's the wrong way to do it. There are extensive foreign commitments for the Canadian military and there have been since (oddly enough) the beginning of the end of colonialism: Suez, Cyprus, Bosnia, Central African Republic, Zaire, Afghanistan, etc.
When I joined the CF, my entry was contingent on the fact that I was going to Afghanistan. You see, the Canadian Forces operate differently than the US Army: you have to volunteer twice to go overseas. Once to enter the military and once for foreign service. A Canadian Foreign Legion could conceivably arise in the mid-50s/60s out of a need to make a force that is consistently at the same strength because every member is available for overseas service.
It would never be a large force. If it got beyond 1 or 2 battalions I'd be very surprised. But that's enough to fill the rolls of a peacekeeping contingent so long as the officers are Canadian, which is why Canadian officers volunteer for overseas deployment most of the time: they need a combat service badge on their record to move up the ladder.
That's just my thought on the matter.