That's impossible to say because not all such boxes are built to the same specifications. Wood species and size, fastener type and schedule, and other factors vary widely. If forced to answer it would be no out of an abundance of concern for safety.

I would follow your plan, but lay 1x4 posts alongside the boxes where appropriate and aesthetically pleasing. Fasten the boxes to the posts and to the bed frame. Make sure it all has enough diagonal bracing, which is the most likely failure concern. That gets into design a fair bit, which isn't really on topic here.

quarter inch pine nailed to 3/4" seems too flimsy to me.

Each time you halve the thickness wood losses three quarters of its stiffness.

If a 2x4 is stiff enough to be a leg for the bed 4 1x4 would work instead or 16 1/2x4 or 64 1/4x4 So at 1/4" thick you need 256" of width to get the equivalent stiffness of a 2x4
your crates have only about 20" so they'll be about 13 times worse.

I'd use solid 3/4" thick wood panels 3/4" (lumber, MDF, or plywood) to construct shelving units to be the legs, if securely fastened together they will be strong enough.