It's been 9 days, but here's an update.
Some specifics on the problem:
When selecting to build None and returning to the main galactic overview screen, the bottom display for the planet correctly shows that it is building nothing.
When selecting to build any ship, no matter the build time shown in the Shipyard screen it shows 0 Weeks Remaining in the overview screen. The next turn, the ship is not built.
Selecting to Purchase the ship does not work, as it drops the time remaining to 0 Weeks again. Overview screen shows 0 Weeks remaining, ship not built. I even tried Purchasing and clicking End Turn while still in the Shipyard screen. The time left did not decrease and the ship did not change.
Oh, and if I Decommission the "broken" shipyard and build another one in its place, or elsewhere on the planet, it won't work.
It's as though the planet becomes officially unable to build ships.
The quasi-good news?
The problem only happens to the existing shipyards.
If I build a shipyard on another planet, it works just fine.
Yes, I think it's a game ruining bug. To be in a full-blown war and suddenly all 20 shipyards stop building can have catastrophic results.
Technically, I can just build more shipyards and I'm fine.
But I'd still file this one as a nasty bug.