That essentially reduces to firing up a terminal and issuing the following command:
sudo apt-get install wine
Repositories are one of the big advantages of Linux over Win. If you need a program, there are chances that it will be in the repos. If so, you can just invoke a clever program which will automagically do everything needed. You can just watch and/or go make yourself a coffee.
Anyway, this is what you do on a normal Linux computer. I don't really know how did the tool you used set up the persistent storage, but I'd bet it will be hard or impossible to make the installation persistent. Certainly there is some voodoo which can be done in order to alleviate the nuisance of installing all the packages into RAM every time you boot the system up, but I doubt it can be done away with completely.
If I were doing things like this, I'd make a custom Debian distribution which would contain the necessary packages, but as I see it, it is probably something you'd prefer not to do. On the other hand, if I (or somebody else) made that custom distribution, it could be packed up in one ISO file and everybody could fashion themselves their own M&M stick easily just with one use of unetbootin (and a little partitioning — but I'm not really sure how would I do that on Win).