Is 'non-work' a word?
According to merriam-webster nonwork is an actual word, wih the exact meaning you just defined. The only differnce is, that it is written together, without the "-" symbol. As for the phrasing of it, for me something like: "Work related only." or "Work related only please." sounds a bit better. I think, anyone using Skype for business would understand that by this you mean that only important, work questions problems will be addressed.