@ martcol
Usually when I come across a page were every JavaScript failing it is because of minification. Not all JavaScript files minify/combine nicely together. I don’t really think the little JavaScript in Connections was the source of the problem.
But to answer your question, yes, you’ll be able to use WP Minify. You’ll have to put a little work into it, which in my experience is always the case but the results are usually worth it.
Judging from the screen captures of WP Minify there not a way to explicitly set which files to minify … only exclude. So is what you’ll have to do is exclude every JavaScript initially, then one by one remove a file from being exclude, clear the minify cache, force refresh the browser and check the browser JavaScript console for any errors. If there are no errors, move on and remove the next one and repeat the process. If there is an error, put the one you just removed back on the exclude list and then remove the next one and repeat the process over.
Hope that helps!