According to R’ Joseph’s opinion that the Israeli army is completely un-Jewish, there should be no need to change anything about the way we go about our daily lives, and no need for any special tefilos.
Funny thing, I saw a picture of some members of this un-Jewish army wearing tefillin. Never saw a goy in tefillin before.
Also, the National Council of Young Israel has an ongoing appeal for sifrei Torah that are needed for the army. They have sent over 150 seforim in the past few years as many small units want to have their own sefer. Now why would a non Jewish army need sifrei Torah?