Halacha brings that one who has no money for Chanukah Lights must sell the shirt off his back to acquire them why is this the case versus almost all other Mitzvos which require up to a fifth of his monies?
The Aruch Hashulchan in Orach Chaim 671:2 notes this difficulty and explains that for mitzvos involving publicizing of a miracle (פרסומי ניסא), including Chanukah candles and the four cups of wine on Pesach (Orach Chaim 472:13), one must spend all of his money.
Of course this just brings up another question: why should פרסומי ניסא cases be so strict?
It says in SA OC Siman 263 Sif 2 that if one doesn't have candles for shabbos he must go around collecting money. The Biur Halacha writes there d'h "al hapasachim" to look into Siman 671 that over there one must be mocher his casus (sell his shirt) for ner channukah. Says the Biur Halacha, so to here by Shabbos. His Makor is the Pri Megeadim in Siman 263 Ashel Avraham Sif-Katan 5. The Pri Megadim however only writes there "u'b'tav-reish-ayin-alef mocher casoso ayin sham". I don't know if he means to say that one must do the same by Shabbos or is simply being m'tzian what one must do by channukah in order to obtain candles. However certainly the Biur Halacha understood that one must sell their shirt in order to buy Shabbos candles.