I don't file in that state, but I'm in CA, and our state e-file is up and running. So the mEF shutdown is only an IRS-level thing. Whether or not your state has e-file shut down or not should be available on your dept of revenue website.
I'd say try to e-file it and see what happens.
MN is a Standalone state (like CA), which means transmission of state returns does not go through the IRS' system, which has been shut down. Technically, you should be able to e-file. When you try (and make sure you check the box to only e-file MN), did you get any diagnostic or error?
Still an AllStar
Don't any states ever shut down their systems to gear up their computers for the new year? Minnesota still processes paper returns pretty quickly so rather than potentially have the thing in limbo until the end of January, is my logic for just paper filing the thing.