Not really. Even with ball point, while for neat letter stroke order doesn't matter, for a messier (normal day-to-day) writing, stroke order absolutely matter. It can become illegible fast if you use wrong stroke order, because correct stroke order tend to create similar overall shape while the wrong order don't. Granted, you don't have to be exact with the stroke order, but general rule still need to be followed -- you can't just write it in reverse order and expect it to be easily readable.

Quite a lot of Japanese people seem to recognise characters less by their shape and more by the stroke connections, in my experience. No matter what the tool used to write it, legibility really requires writing the characters in the correct order for the most part.

I know when my dad grew up, in Scotland, he'd get hit with a ruler if he was seen writing with his left-hand..

Happily the rules had relaxed by the time I had to write left-handed, and very messily with a fountain-pen!