My DH believes the same thing - when he was dieting several years ago he had 'free weekends', now it is one day per weekend. Typically though, he has one 'cheat meal' that is high calorie enough that he is too full for too much else later in the day. When I first started calorie counting I did it with him, though still tracked my calories for the day (he doesn't track at all that day.) That one day brought my daily average up by 150-250 calories some weeks. So, I started cutting back more and more on those days and just not talking about it.
That being said, I do still have a higher meal every week than a normal meal, but still track it. I also plan in some higher calorie snacks and stuff here and there. I just had to let go of the cheat day concept.
I should add, DH has lost and regained and lost and regained up to a 75 pound swing over the last several years. This time he lost 20, regained 10, is working on losing again. So, could be that the 'cheat day' set him up for long term failure. I learned how to count pretty much anything I want into my calories for the day, or even for the week sometimes if I go higher than planned, so learned not to binge. DH hasn't learned that yet.