Well, the answer is yes to both. Yes you can provide your own box OR you can use the boxes USPS provides--which, confusingly, say Priority Mail on them. They recently (within the past year or so) began allowing use of Priority Mail boxes for express shipments internationally; this happened when "surface" international shipments stopped being offered.
I have to correct what I said!!! They DO have specific Express boxes, which are free, but you can NOT use Priority boxes for Global Express. Sorry for the misinformation! I was only half on the right page there.