I consider the question "What's the smallest sample size for which a box-and-whiskers plot is a useful visual summary" to be about a rule-of-thumb for making good plots. (The question "Should implementations of a box-and-whiskers plot enforce a minimum sample size" does seem to be about opinions rather than practice.)

I looked for advice in a few books about statistical graphics. It seems straight advice is hard to find. So far I have:

[1] J. Tukey. Exploratory Data Analysis (1977)

First a general comment on page 29:

If we are to select a few easily-found numbers to tell something about a batch as a whole, (...) we would like these values to be easy to find and write down, whether the total count of the batch is 8, or 82, or 208.

And more specifically about boxplots, which Tukey calls schematic plots, in reference to visualizing 15 weight measurements from a 1893-94 experiment by Lord Rayleigh:

Here the main issue (...) is made quite clear by the individual values of the dot plot--and almost completely covered by the schematic plot. (Only almost, because the experienced viewer--finding the whiskers so short, in comparison with the box length--is likely to become suspicious that he should see more detail.)
Clearly we cannot rely on schematic plots to call our attention to structure near the center of the batch (...)
Exhibit 11 uses the schematic plots for one of the purposes for which they are best fitted: comparison of two or more batches. In it, the two batches of Rayleigh's weights (one batch of 7 from air and another batch of 8 from other sources) are set out and compared.

[2] F. J. Anscombe. Graphs in statistical analysis. The American Statistician, 27(1):17–21, 1973.

Each datasets in the famous quartet has 11 points, so Anscombe's implicit advice is to not summarize fewer than 12 points?

Summary: John Tukey suggests indirectly to have at least 8 points for a box-and-whishers plot.

I consider the question "What's the smallest sample size for which a box-and-whiskers plot is a useful visual summary" to be about a rule-of-thumb for making good plots. (The question "Should implementations of a box-and-whiskers plot enforce a minimum sample size" does seem to be about opinions rather than practice.)

I looked for advice in a few books about statistical graphics. It seems straight advice is hard to find. So far I have:

[1] J. Tukey. Exploratory Data Analysis (1977)

First a general comment on page 29:

If we are to select a few easily-found numbers to tell something about a batch as a whole, (...) we would like these values to be easy to find and write down, whether the total count of the batch is 8, or 82, or 208.

And more specifically about boxplots, which Tukey calls schematic plots, in reference to visualizing 15 weight measurements from a 1893-94 experiment by Lord Rayleigh:

Here the main issue (...) is made quite clear by the individual values of the dot plot--and almost completely covered by the schematic plot. (Only almost, because the experienced viewer--finding the whiskers so short, in comparison with the box length--is likely to become suspicious that he should see more detail.)
Clearly we cannot rely on schematic plots to call our attention to structure near the center of the batch (...)
Exhibit 11 uses the schematic plots for one of the purposes for which they are best fitted: comparison of two or more batches. In it, the two batches of Rayleigh's weights (one batch of 7 from air and another batch of 8 from other sources) are set out and compared.

[2] F. J. Anscombe. Graphs in statistical analysis. The American Statistician, 27(1):17–21, 1973.

Each datasets in the famous quartet has 11 points, so Anscombe's implicit advice is to not summarize fewer than 12 points?

Summary: John Tukey suggests indirectly to have at least 8 points for a box-and-whishers plot. He also has a hint about catching out a misapplied boxplot.