In Major League Baseball (not HBD), if a team trades a minor league player without options to another team, is the acquiring team required to waive them before assigning them back to the minors?

I don't think so. Players like that only need to clear waivers (or get DFA'ed) when they're being moved from the majors to the minors, not moving from one minor league club to another.

If that's true, why does HBD require it?

Anyone know why HBD requires minor league players without options to be waived again (to go back to minors) after completing a trade?

Because that is how it works in MLB. If a player who is traded is out of options, he must clear waivers before being placed in the minors. If he was never on the 40, he of course never burnt through his options and can be placed at any level.  He is not "moving from one minor league club to another" as the previous poster said, he is being traded from one franchise to another franchise. This is an important distinction - trades are between franchises, not minor league levels. HBD is consistent with MLB.

It's the same reason why players that are out of options can't start the year in AAA in real life.  You can't be assigned to a minor league team without passing through waivers first.