Can my 12 year old cross with just his birth certificate?

Yes and a school ID would be very helpful as well. They will also ask him some simple questions such as school he goes to, his address, parents names and name of city of residence.

A valid passport is required.

My son is 7 years old, can he go through CBX with his American birth certificate? He would come with his mother who is a resident

I think yes, they are going to ask you questions and possibly tell you to bring your passport next time