The only walkthrough I could locate for this game suggests that the way to get through the swamp/bog is to first eliminate all enemies then to follow the exact path taken by the German Shepard through the bog.
Unfortunately, the dog just stands there, doing nothing.
Is there something that needs to be done to "activate" him?
Is there a map anywhere showing the path? Thanks.

Have you tried shooting near him to see if that gets him moving?

Shooting and/or punching the dog has no other effect than to reduce his health bar.
Saving and reloading also does not shake him loose.
I strongly suspect he runs his route right at the very exact second that
the last enemy is killed----- and does so in a fashion that is deliberately designed
to be too quick to scroll back, watch, and memorize.
In some reloads, he stands around and does nothing. In many, he runs off to some unknown place, certainly not to the goal across the bog.
I'd like to find the game designers and choke them to death.
This is the exact thing that makes older games suck so bad......the AI isn't very good, so they deliberately design in something frustrating and obnoxious like this.

Here is what I located online, and a direct link to the Crash Site walkthrough (Section 2.03).
You have made it further than I have as this game can be very unforgiving and one that seems to take many replays to get it "right."