You can always tack up bits of scrap wood to wedge your bed in place while you screw it together. Or use vehicle cargo tie-downs as band clamps. (This one time at band clamp... ) Do you have any woodworking tools, like c-clamps? It is flea market season around here anyway, and that is a CLASSIC flea market item- crappy made-in-China clamps and bungee cords.
Rubbermaid tubs will work too. Are you planning on putting in drain holes? You should lay down a thick tarp under the milk crates to protect the floor either way. Preferably with the edges wrapped around 2x4s so that it's like a catch basin.