He wants, and is getting, attention for this. I agree that ignoring him - literally walking away and saying you don't listen to screaming - works best. Some kids need immediate response - so taking away TV later on doesn't work. So you put the game away if he loses and doesn't like it. You also let kids win some of the time, as long as they don't know it. But you can find other games - and that's what you say, "Okay, I'm putting this away since you don't like to play." Then you walk away and tell him you'll come back when he calms down. Or you send him to his room, not with a lot of yelling and conversation, but "until he feels calmer" and wants to return to family fun.
Having kids with 2 households is very difficult - you want to maintain your standards/values, but it's not consistent with the other home/parenting style. Transitions are very hard for all kids of divorce, and impossible for some kids, and that's what you're seeing.
I know you don't believe in medicating, but do you have any legal rights? Probably not - this is a decision for the father and mother to make, right? What is their belief and what, if anything have they tried and discussed? If they don't want to medicate, that's their decision. I work with plenty of families who don't medicate, so I support that, but as a stepmother myself I am telling you that you can't make this your decision.
I'm not surprised that you aren't getting too far sitting down and talking to him about this. He's 7. He doesn't have the vocabulary. He probably doesn't know why he's doing what he's doing. Very rarely will a parent of a 7 year old get him "to say too much" - and it's harder when you are the stepparent because he knows he has 2 other parents to deal with.
I'm particularly disturbed by your comment that he's about to be kicked out of school? What??? Is it a private school? If it's a public school, they can't kick a child out unless there are persistent disciplinary issues that have not been and cannot be death with - usually kids are only expelled at older ages, like teenagers. Your husband and his ex should be sitting down with the teacher, the school psychologist, and any other resource personnel to do a plan that involves classroom observation, immediate intervention strategies, the assignment of a paraprofessional aide and perhaps the development of an IEP or other plan. I taught in 2 different private schools, and we also used psychologists and aides to support students with behavioral and learning issues. Even if there is no diagnosed learning issue, if there's a behavioral problem, it becomes a learning issue.
There should be a consistent strategy and a consistent vocabulary among the school and the 2 homes. Kids do much better when there aren't 3 sets of rules and expectations. I think parents getting into family counseling can really help to develop a parenting plan. I'd also look at what happened to trigger a resurgence of this behavior. A qualified professional can help you do that. Your husband and his ex should find a family counselor who specializes in children - the child's pediatrician can make a referral to someone who accepts the family medical insurance. The child's insurance can be used for any sessions the child is part of, even if just for part of the session, and the adult insurance can be used for other sessions - so that allows for a lot of coverage.
Please give this kid a chance with some good. solid intervention. He needs to evaluated medically, psychologically, neurologically, socially. He's in pain - he doesn't want to be like this. But it's often not something that one parent, particularly a stepparent, can talk him out of.