I freeze DD's drinks all the time - when I use juice boxes I take them out of the freezer first thing in the morning when I get up and drop them into the insulated lunch bag just as we're about to leave. You could always place the frozen juice box in a bowl of room temperature water to kick-start the thawing.
More often than not, I freeze her juice in a reusable bottle (Rubbermaid makes quite a variety) Usually, I'll half-fill the bottle and freeze that. In the morning I then top it up - that helps to speed up the thawing!
The upside of packing a frozen juice bottle or box is that you won't need to buy a separate ice-pack to keep things like tuna cool!
