As the title says, are Hitbox controllers legal to use as long as the inputs follow the rules (no left+right or up+down)? Or are only traditional d-pads allowed?

Hitbox uses an SOCD cleaner which is a device that makes left+right=neutral and up+down=up.

For reference I am specifically asking about this controller: https://www.hitboxarcade.com/pages/hit-box

I do not own a Smashbox, but I am wondering about the ruling for that controller too.

The general rule of thumb is this:

If the controller you're using doesn't provide advantages (L+R, turbo, etc.) over an original, unmodified NES controller, then it should be fine.