For wine, the white styrofoam packaging that acts as an insert in a cardboard box is my strong suggestion - I've brought back the spoils of wine tours from all over the world in these with a huge amount of success. I also recommend writing WINE on the side of the box - baggage handlers see FRAGILE all day long on a variety of things that may or may not be, but I've found that actually declaring the contents of the package helps handlers make well-informed decisions about how to handle it and most don't want to deal with cleaning up a leaky box

Back on topic, for the plastic bucket, these can be checked and will likely arrive at the destination, but don't expect the lid to hold and don't expect them to be reusable. Every time I've done this, the plastic bin arrives shattered - there's usually at least one crack in the body of the bin, and the rim has lost random shards of plastic making the lid no longer be retained. If you're trying to move a bunch of linens or something, this may not be a big deal, and sacrificing a $7.99 box to move a large quantity may be a reasonable tradeoff. I find that a cardboard box is a better receptacle in most cases, though.