A person with a Y chromosome develops into a male because of the
presence of a gene on the Y chromosome called SRY (sex-determining
region of the Y)gene.  What would happen if an X chromosome, by
chance, has this gene in an XX individual?  Is it a true statement,
then, that only XY individuals can be males?

XXY (Klinefelter's Syndrome) and XYY (Jacob's Syndrome) are
established conditions. A Reseacher can give further information.

SRY gene, produces tdf, testes determing factor.  

If an individual has SRY on an X chromosome, they would mature as males.

If an individual has a defect in the SRY gene, they would mature as females.

This is the earliest part of the sex-differentiation pathway, and thus
both genitals and brain would develop  in either a masculine (+SRY) or
femanine (-SRY) direction.

Testes will develop in the presence of tdf, testes will produce AMH
(anti-mullerian hormone) which blocks maturation of the mullarian
ducts (primoridal female internal reproductive tract), and thus the
wolfian system develops. Testosterone (more specifically, a metabolite
of testosterone, dihydrotestosterone) masculinizes the external
genetalia.

In terms of brain differentiation, again testosterone is responsible.
Testosterone is converted by aromatase to estrogen which then is
responsible for brain sexual dimorphisms.

Estrogen is not sufficient, as maternal estrogens are high, and are
bound by alpha-fetoprotein to prevent masculinization when
inappropriate.  Thus, T, which is converted to E in the brain of the
fetus is needed.

That is a really brief, and cursary summary.

I also wanted to add, 

"only XY individuals can be males?"

It depends on your definition of male.

Genetically, only XY individuals are male.
You can also look at the internal, external, secondary sexual, and
brain characteristics.

I think it really has to depend on what level of analysis you want to look at.

And, for the record, it does happen that occasionally during meiosis a
translocation of SRY to the X chromosome does occur.