Q. How do I cite in MLA 7 a book sectioned like a journal (e.g June 14, 1968 [entry], September 19 1968 [entry])?

I am a bit confused with what you are up to. Here is what I think you have ...

a).  A book that is written by one author (as opposed to an edited book where there is a general editor for the work and each chapter is written by a different author).

b). In this book, instead of headings such as "chapter 1" the headings are dates, such as "June 14, 1968."

If this is the case, then I would treat it like a book, and not a book with chapters. (and I assume the pagination runs continually through the book and does not start again with each chapter?).

If the dates/chapter headings give context to what you are writing, then you could add that in the text of your sentence.

In November of 2016 White writes "teaching your Australian Shepherd to sit on his back legs and cross his front paws over his nose is one of the worst things to do if you wish to enjoy a peaceful dinner" (67).