Well after an hour and a half of cursing Tableau's name I figured it out. The problem was that Tableau doesn't make it obvious how to do a simple plot of points. I had selected "Dimension" for my row data type, as otherwise it was only giving me an aggregate value like sum or average--not very useful for a box plot. To get it to plot actual points, including duplicates, you have to select the Analysis menu item and deselect "Aggregate Measures".
This is confusing, as it appears to mean that it is all or nothing with your aggregate functions, and that you're deciding at the worksheet level rather than the data series level if you're going to be using aggregates.