I was expecting this. I suppose there are victims of the American public school system who don't know about the Nazis. Ah, but there have been all sorts of colors associated with shirts worn by members of dubious organizations. Black Shirts, Green Shirts, Blue Shirts, Silver Shirts, Yellow Shirts, and, of course, Black Shirts.
When you give commands, you're using the verb form that's known as the "imperative". Spanish has two flavours of that, one for positive commands ("Do that!"), and one for negative commands ("Don't do that!"). They are different for some subjects (but not all), and the negative command form, which is what you'd use in this sentence, is always the same as the subjunctive verb form.
For this sentence, the negative tú command of ponerse is "te pongas":
- Nunca te pongas la camisa marrón. - Never put on the brown shirt.
Do yourself a favour and step outside of Duo for a bit, if you get bogged down... works for me.
There are plenty of Spanish language learning options.
T.V., YouTube, instructional materials, library books, clubs etc; which can be used to keep it fresh and continue making progress, or simply to access another teaching style.
The shirt is an object in this sentence, though. But yes, that's approximately the reason here. Poner usually needs another argument, you put something (DO) into some place/condition/state (IO). English uses reflexive verbs much less often and mostly opts for using intransitive verbs (or in this case, verbs with a reduced object count).
Ponerse does mean “put on” but it is an infinitive verb, so it needs a conjugated verb to precede it. For instance: Ella quiere ponerse la camisa. She wants to put on the shirt. Tu quieres ponerte la camisa. You want to put on the shirt.
In the sentence given, pones is the conjugated verb, and to make it reflexive, the te is needed in the sentence. But you can’t add the te to the end of a conjugated verb; it needs to go before the verb.
You cannot attach object pronouns to the end of a conjugated verb (unless it's conjugated in imperative, the command form). The only other verb forms you can attach the pronoun to are the infinitive form ("No puedes ponerte esa camisa") and the gerundio ("Saliste de la habitación poniéndote tu camisa").