Is it possible to check in plastic storage boxes on an Air Canada international flight?

You are flying FROM New York, hence you are at the mercy of TSA. And if the scanners show anything suspicious looking in your boxes, they will open them to inspect, in which case your duct tape, plastic wrap, etc will be cut open and useless.

What I usually do with boxes and cases like that is to get some compression straps for backpacking that are long enough to wrap around in either dimension and then strap the box shut (with straps going both directions and buckles on top of the lid). I then use duct tape to wrap around the case over the compression straps just below the lid. That way when TSA undoes the straps, they are right there ready to be refastened. I also tape a polite note, under where the buckles are, addressed to TSA, 'thanking them for their service protecting we travelers and asking them to be sure the straps are tightened down again' (not the exact wording ;-). It comes through just fine and I am often flying much further internationally.

This doesn't really address theft, but realistically when you consider how many bags get checked every day versus the number of actual thefts, air travel is relatively safe. And common sense says not to pack anything of value in a storage tote that is being checked.

Be aware that baggage handling systems are known not to be too gentle. It could pretty happily put your box upside down in a cart on a steep incline while heading to another part of the building at 30 km/h. When it gets there, it is usually tipped off the cart at speed and down a steep chute to the waiting luggage handlers below.

If my impression of your plastic box is correct (I'm thinking transparent IKEA ones), there are plenty of protruding edges and sharp corners it could get caught on. I'd advise using plenty of duct tape or IMHO taking cardboard boxes that are less likely to crack, better at absorbing impact and flush on the sides. I'd advise putting the heavy stuff at the bottom to encourage it to stay upright as desired if it is tall.

I can't possibly comment on what you're likely to experience with the TSA, however I can recommend good storage bin for air travel.

Before moving to the UK from Canada in 2010, I bought two 24 gallon Rubbermaid ActionPackers and a 4 pack of padlocks (all with the same key). I filled them both with my hand tools that I couldn't bear to leave behind, and also a couple blankets or pillows or something to keep things from bashing around too much.

I brought one bin and my parents brought the other a few weeks later. We flew the same route, Air Canada to YYZ and then IcelandAir to LHR via KEF.

Both bins and contents survived perfectly intact. None of the locks were cut by security in any of the three countries we passed through. My parents and I had keys on us at all times just in case we were required to open the bins, but nothing came of it. The only friction in the entire process was being asked to bring it to the oversized baggage drop. They x-rayed it in front of us and asked what was in it, but that was all.

The ActionPackers are a bit pricey compared to other similar sized bins but they're sturdy and lockable, plus you'll end up with a good quality bin at the end of it.

If you're worried about the TSA opening it up, you might be able to use the TSA approved locks, however you'd have to test this yourself as I don't have any TSA locks on me. I'm not sure of the minimum lock throat depth the ActionPackers require either.