AviK: The Sridei Eish says Shirei Eretz Yisrael? I thought his Lashon has “Shiros V’Sishbachos LaHKBH”, but I could be wrong. Either way, whatever the army women were singing I don’t think qualifies. I don’t think their intentions were to sing L’sheim Shamayim. And I won’t even begin to discuss your brutalization of the Sugya of healing a Nochri on Shabbos, or the fact that the comparison is entirely inapplicable..
Rav Aviner makes a very good point about EE Efshar V’lo Kam’chaven.