It sounds like you and your son had a traumatic experience.

I am not sure what happened to you but it seems to me that you were "denied entry" rather than "deported."

Getting deported happens, for example, when people overstay their visas or get caught in violation of the terms of their visas. In such cases, they are typically prohibited from entering to Japan for some time. If you were denied entry because of an error in documentation you may find that you are free to enter Japan when you have satisfactory documentation. As of today (October 11) there is no longer a visa requirement, so the documentation requirements are minimal. Of course, you should check with a Japanese embassy.

Speaking from personal experience of having been denied entry to Canada over 20 years ago - It has not prevented me from visiting and even working there, but it does mean that I get called for secondary interview when I arrive. That's inconvenient and time consuming but not disastrous.

I think that there are two points that you need to deal with:

1) Establish what happened and what was at fault with your documentation
You need to establish what was wrong with the documentation that you had.
It's possible that the agent made an error in the application. In that case, I think it's likely that you can get some compensation from the agent. (I am not a lawyer).
It's also possible that the agent issued you with the certificate for ERFS registration, which you were required to take to a Japanese embassy to get a visa issued. In that case it may be difficult to establish who is at fault. Were you instructed to get on the plane with just the certificate?

2) Establish whether you are subject to any restriction on entry to Japan
First of all, I think you need to clear up the confusion about what happened (see point 1 above) and then discuss the matter with a Japanese embassy or consulate. I presume that you received some paperwork when you were sent back to the US? Keep it all to document what happened,.

I hope that you can recover quickly from the trauma and that you soon get clarity on what went wrong and what you can do about it. I feel optimistic that you will be able to visit Japan again soon.