I think you are looking for the xdiscrete_values keyword, e.g.:
using DataFrames, StatsPlots
df= DataFrame(cat = [16384, 32768, 65536, 131072, 16384, 32768, 65536, 131072, 16384, 32768, 65536, 131072], val = rand(12))
boxplot(string.(df.cat), df.val, xdiscrete_values=string.(df.cat))