What is a gender neutral equivalent for the following phrase?

Try this:

I'm not going to get all bent out of shape.

No gender implied at all, and the meaning is the same (to get upset to distraction about something).

1. (of a person) Upset or angry.
   Don't get all bent out of shape—I'm sure she didn't mean to insult you.
   You should apologize to Phil before he gets bent out of shape.
   _{TFD Online}

There's nothing wrong with "... get your knickers in a twist". It's the idiomatic expression, and it can be applied to anybody regardless of gender or underwear choice. They are, after all, only metaphorical knickers, not literal ones.

Replacing the nouns in that expression will just make it seem as if you don't know the idiomatic form. Whatever gender-neutral boxes it might tick, it would fail to be good English.