Hard to say how to do this without a working example. My first guess would be to just add an integer column with the orders that you want.

A simple, brute-force way would be to add each boxplot one at a time.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

df = pd.DataFrame(np.random.rand(37,4), columns=list('ABCD'))
columns_my_order = ['C', 'A', 'D', 'B']
fig, ax = plt.subplots()
for position, column in enumerate(columns_my_order):
    ax.boxplot(df[column], positions=[position])

ax.set_xticks(range(position+1))
ax.set_xticklabels(columns_my_order)
ax.set_xlim(xmin=-0.5)
plt.show()