While I am not familiar with the details of the proof, I think the example being referred to is given by taking the algebra of bounded Baire functions on the real line, and quotienting out by the ideal of functions which vanish on a nowhere dense set. Historically this was of note because it is an example of an $AW^\ast$-algebra in the sense of Kaplansky that is not a von Neumann algebra.

If you look at the last sentence of the MathReview which says

Also, a stonian space is characterised as the spectrum of the totality of the self-adjoint algebra of the totality of bounded Borel functions (modulo those vanishing on a set of first category) on a Hausdorff space. In this way, an example is given of a stonian space which is not hyperstonian. then this would seem to bear out my hazy recollection.

I have a feeling that the example can be found either in Dixmier's C*-algebra book or in his von Neumann algebra book - admittedly that is not too helpful given the density of those texts, but you might be able to find it. See also an article by Kadison:

MR0861016 (88f:46114) Kadison, Richard V.(1-PA) The von Neumann algebra characterization theorems. Exposition. Math. 3 (1985), no. 3, 193–227.

which apparently describes the Dixmier example (I don't have electronic access to the paper).

While I am not familiar with the details of the proof, I think the example being referred to is given by taking the algebra of bounded Baire functions on the real line, and quotienting out by the ideal of functions which vanish on a nowhere dense set. Historically this was of note because it is an example of an $AW^\ast$-algebra in the sense of Kaplansky that is not a von Neumann algebra.

If you look at the last sentence of the MathReview which says

Also, a stonian space is characterised as the spectrum of the totality of the self-adjoint algebra of the totality of bounded Borel functions (modulo those vanishing on a set of first category) on a Hausdorff space. In this way, an example is given of a stonian space which is not hyperstonian.

then this would seem to bear out my hazy recollection.

I have a feeling that the example can be found either in Dixmier's C*-algebra book or in his von Neumann algebra book - admittedly that is not too helpful given the density of those texts, but you might be able to find it. See also an article by Kadison:

MR0861016 (88f:46114) Kadison, Richard V.(1-PA) The von Neumann algebra characterization theorems. Exposition. Math. 3 (1985), no. 3, 193–227.

which apparently describes the Dixmier example (I don't have electronic access to the paper).