My understanding of Stillman's is that is it a severe form of ketogenic diet and is not easily sustainable for longterm. Basically, it is used to break plateau's.
Lean meats (all fat trimmed), Chicken & Turkey, as well as fish are allowed. No butter, margarine or oils are allowed. Eggs are allowed, preferably hard boiled. Cottage cheese, farmer cheese & pot cheese are permitted, however, must be made with skim milk....not whole milk.
You might be able to do this for a few days...maybe even a week...but, it would be difficult to follow long term. In my opinion, Atkins Induction is much more feasible. Dairy can be "touchy" for some. Personally, I noticed when I cut back on the cheese, the scale started to move. I'm on maintainace now and I have added 4oz. of low carb/low sugar yogurt every morning and it has not affected my weight loss. Everyone is different though and sometimes it takes trial & error to find what works best for you.
All the best to you!!