Simple Mendelian sex-linked chromosome fruit fly question
I think the text is slightly misleading. But before going into the detail genetics of eye color let's answer your first question.

First, wouldn't recombination scramble the eye color alleles among the sex chromosomes?

Two X chromosomes do recombine but one X and one Y chromosome do not recombine (to the exception of the so-called pseudo'autosomal region).

I am not sure why recombination would change anything here. I think the eye color of flies in determined by a single locus.

How is this possible?

Let's go through the text slowly because I think the text is a little misleading. I am suggesting the only scenario I could think of that would explain the observed pattern. Note that this scenario also depends on the interpretation of the text and I would need more info on the experience to make sure my interpretation is right.


  When he mated red-eyed flies (dominant) with white-eyed flies (recessive), most, but not all, of the F1 progeny were red eyed.


Actually, only males are sometimes white eyed here. And to be more accurate, if sex was uncorrelated form eye color in the parental lines, then half of the males should have white-eyed.

What was happening here is that red is dominant, white is recessive and the locus for eye color exists only on the X chromosome. As a consequence, all F1 females are now heterozygous (and display the dominant red-eyed phenotype) and all males are hemizygous, half of them having the red-eyed phenotype while the other half have the white-eyed phenotype.


  Furthermore, when Morgan mated the red-eyed males of the F1 generation with their red-eyed sisters, [..]


Let's forget about the white-eyed males from above, we will just cross the red-eyed males with their sisters.


  they produced about one-quarter white-eyed males, but no white-eyed females.


By they produced about one-quarter white-eyed males what is meant is that a quarter of all individuals were white-eyed males. That is half of the males were white-eyed. The other three quarters of the individuals were red-eyed.

The males received their X chromosome from their mother only (as they can only receive the Y chromosome from their father). Half of the males have therefore received the white-eyed allele while the other half have recevied the red-eyed allele. For the females, they all received from their mother either a white-eyed allele from their mother or a red-eyed allele. As we selected only father that had red eyes in the previous step, all father carry exclusively the red-eyed allele. As a consequence, all females receive a red-eyed allele from their father. Therefore half the females are homozygous red-eyed or heterozygous. In any case, all females are red-eyed.