This is happening to everyone today. I was playing earlier and had the problem myself (with two different characters). I also saw at least 5 people in various sectors asking after the problem too. It's never been this common-an issue, least not that I've noticed.
How I was told to fix the issue (and how I told others in-game) is as follows; cycle through your Bridge Officers using the Universal Slot that comes with your Shuttle. Select one Bridge Officer at a time there, and then select them in the actual Boff Slot you want them to be in. Then go back to the Shuttle Slot and select the next (etc etc).
I had the same problem, I could not assign any of my bridge officers to their slots. I reached level 50 for both my engineering and science toon and had new ships. I played the second wave mission for both of them and during the mission I was able to assign bridge officers to their stations. I have no idea if going on the other missions would do the trick, but playing second wave worked for me.