As I previously suspected, this article states that you can only leave a group chat if ALL participants are using iMessage on iOS 8 or higher:
http://www.imore.com/how-quickly-leave-group-messages-imessage-ios-8
I would check all your group chats again. The 'Leave Group Chat' option was not in iOS 7 so for "Mike, John, Steve, and Eddie", they will definitely not get this option, and I suspect that you won't get this option either.
Perhaps the group chat you have with "the same Mike, John, and Steve, but not Eddie" which you can't leave, might have some standard carrier sms messages within that whole threaded chat - that is why you cannot leave them. The whole thread (between everybody in the group) may possibly have to contain ONLY iMessages (no sms carrier messages whatsoever).
Nope. Both groups are using iMessage. Also, the option to leave was even present in red when I was running iOS 8 betas, and all of the other group members were running iOS 7. I can confirm that some of the members of the group are running iOS 8, but not all of them.
Groups using SMS don't even give you the greyed out option to leave the group. The option just isn't present at all.
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So you're saying 4 people in the group plus yourself or 4 total? I have only been in group chats that large with at least one person on an android phone. I really need to test this with a group of iPhone users.
In groups where there are 4 TOTAL members (including myself), the option is in red and clickable. In groups with less than 4 (including myself), the option is greyed out. And just for added confirmation. The 4 person group contains A, B, C and myself (and is red and clickable). The three person group contains the same people except C, and is not red, nor clickable. Both groups are 100% using imessage.