First, you may want to post your own new thread (rather than posting in one started by someone else) to get more attention from other posters; however, this is definitely the right forum.
Dog aggression and human aggression are not related to each other at all, so it makes sense that he is fine with people but aggressive with dogs.
In my opinion, neutering does often help with aggression, but it won't fix it, particularly since he has had time to practice this behavior. Neutering him does not sound like a bad idea at all, but it probably won't be a magic bullet. GSDs also may just naturally be dog aggressive; it depends on the breeding and the individual, but it is something that is found in the breed. In those cases, your dog may never be good around other dogs. I suspect that he naturally has some dog aggression, and that he also has a strong prey drive which is what prompts him to go after squirrels and cats.
Are you responsible for walking this dog, or does your boyfriend take responsibility? If you cannot control him, you should not be taking him outside. It doesn't take strength to control a dog if you set him up correctly with training, but you don't know how to do that. However, if your boyfriend is taking him out and just relying on strength to keep him in check, your dog is only going to get worse.
IMO, you need to get a trainer involved ASAP. This is a very dangerous behavior. Even though he is not human aggressive, someone may get bit trying to save their dog, and you could be sued and/or have your dog put down. It also isn't fair to anyone else who happens to be walking their dog when you take him out. This is not something to muddle around with on your own.
While you are looking for a trainer, you need to muzzle him any time he is going to be around other dogs. This is for everyone's safety. He should also never be off-leash around other dogs (even with a muzzle) and you may actually want to confine him to the house and yard for the time being, except taking him to vet and training appointments. Every time he practices this behavior, it reinforces it.
I doubt he "immediately" goes over threshold. Most likely, there is a distance where he is controllable, but you are not noticing early signs that he is about to go over threshold. It's easy to do, those signs can be very subtle. Most likely, your trainer is going to work with you to teach him to walk nicely on the leash at a safe distance, then gradually increase distractions. The primary goal should be showing him what you want and rewarding him extensively for it (so that walking calmly is more rewarding than lunging and attacking), and only correcting him if absolutely necessary.
For a good illustration of this, watch some episodes of "It's Me or the Dog" with Victoria Stilwell. She uses positive methods and I've seen her deal with a lot of leash reactive or aggressive dogs. This is NOT a replacement for going to work a trainer, but it can help you visualize what I have found to be a very effective method for these kind of issues.
And remember, your dog may never be trustworthy around other dogs. Some dogs just aren't, it's their personality. However, I've never met a dog with a decent temperament who can't learn to at least walk quietly by other dogs in public.
edit: you also could try posting your own thread in the "Aggression" forum under this same section. I thought that's where we were.