Ok here goes a translation from wikipedia-jp on the origin of the word osu. There are two separate meanings - one which was used as a greeting between upper and lower students in some insitutes in japan. Both meanings are very deep in history and not used (with the kanji seen in the Ouendan title) in recent years. The second meaning has more relevance to what you would be interested in though:
Theory has it that in some wards in Japan (can't be bothered translating place names, irrelevant) where the samurai was born, clans used very servere discipline on their members. It is said that the young members used to shout "OSU!" when seeing/meeting each other in the chaos, trying to cheer each other up. It is thought that the "osu!" that cheer groups/martial artists use was born from this meaning, moreso than the previously mentioned "osu" as a simple greeting (which is used in a more plain context of saying "hello").
Even in today's japanese culture, osu is used as a greeting (regularly heard as "ossu!" between friends) and is actually quite popular amongst the youth. But as you probably gathered from the translation above, this has a very different meaning to what is used by ouendan - and is never seen using the "押忍" kanji, but simply in hiragana or katakana "オッス！".
ohayou gozaimasu (good morning)
The origin of the kanji is as follows (in its original martial arts/ouendan-ish form):
(押して)oshite (忍ぶ)shinobu (to push and withstand) - having the meaning of "swallowing one's ego and bearing with it"
Leads us to the final 「押忍」.
Hopefully I'm not too far off with this translation