It is inappropriate to wear an IDF sweatshirt, regardless of how naive its wearers may be.
Every army indoctrinates its members in its nation’s core values. Regardless of your views on the State of Israel, and how much Zionist kool-aid you’ve had regarding how the IDF protects Jews, etc. ad nauseum, those core IDF/Zionist/anti-Torah values, including the rampant znus there, and that the gedolim have already ruled it is Yehareig viAl Yaavor to join that army, are all obviously very much not in consonance with those of a BY Girl.
I’m surprised this is even a hava amina. Besides for the whole Zionist mess, the gedolim said it’s better to forfeit one’s life than to join the IDF. Yet some BY girls think it’s okay to wear a sweatshirt with the IDF logo imprinted on it because they want a wearable memento from Eretz Yisrael?
(Would you wear a sweatshirt with an image of an avoda zara on it as a memento from a trip to Rome, lihavdil?)
Do they not make sweatshirts with Yerushalayim or the Kosel or something else that’s actually Jewish, if you must have a memento from E”Y? Even if not, where is your sensitivity?
On another note, who said you have to support the army, and who told you it’s a kiddush Hashem to show the nations you support the IDF? Maybe you have no business supporting that which the gedolim ruled is yehareig viAl Yaavor to join and that it’s in fact a chilul Hashem to show the nations that you support anyone but your host country?
Ask your LOR, but Zionism is a very big problem, even if the current attitude to it is essentially, “it’s terrible that Zionism came around, but now that it did, we can’t do anything about it”. This doesn’t kasher their various forms of rishus and it does not kasher their State or anything else about them, either.