Unlike most British Quakers, Nixon was not a Gurneyite Quaker. There are a number of Quaker flavors, not all of them pacifist. As I understand it, the sine qua non of Quakerism is following the inner light, not any particular doctrine.

Nevertheless, when questioned about his war service, Nixon claimed to consider himself a pacifist due to his "Quaker heritage."

Here is an interesting account of Nixon's Whittier Quaker meeting and the role that "heritage" played in Nixon's self-presentation:

https://www.friendsjournal.org/richard-nixons-first-cover-up-2/

At any rate, his service as a de facto non-combatant in WWII pales as an impeachment of his Quakerism beside his carpet bombing of Southeast Asia.