They do not receive suppression tokens, so hitting the dirt would not grant them any cover correct?

I'm of two minds of this, not sure which is more accurate.

1. They would be able to hit the dirt as normal, 'receive' the suppression marker but ignores it.

2. Because its worded that it follows the mechanic in the suppression rules, it has a prerequisite of having to be able to accept a suppression marker, which means no they can't use it.

Good question! My gut reaction would be #2

On page 53 of the rule book in chapter 3: spec ops under "Blutkreuz Ape" reads as follows- "Ignore all effects of Suppression markers; NEVER place Suppression markers on a unit with the Blutkreuz Ape ability. This means that they DO NOT gain the benefits of the "Hit the Dirt!" reaction. Hope that helps!