The part they don't like is probably the general feeling that this will devalue their product. Ticket prices already "feel" high to the general public. Having advertised services out there (even if ultimately untenable) that drastically undercut standard prices will not help this perception at all.
I don't understand their complaint other than, they didn't think of it first.
MoviePass is dropping their pricing to $10/month, this screams out "Hail Mary" to me. I wouldn't be surprised if MoviePass was on its last legs financially (they just sold a majority stake, see http://variety.com/2017/film/news/moviepass-1202527956/) and their contracts with the chains could potentially leave the chains with millions of dollars of owed ticket money if they suddenly stop operating.
If that's the case, no wonder AMC are trying to get out of it.