From One of Rabbi Edley's articles:
In the Mechilta (Shemot 20; 8) (The commandment stating to "Remember the day of Shabbat") it states:
Rabbi Yitzchak says, ‘Don’t count like the
others count, rather you should count for
The explanation is that non-Jews count the
days of the week with names, each day has
its own name, either names of heavenly
bodies [or gods] (Sunday, Monday etc.) like
the Christians do, or other names. However
Israel count each day with reference to
Shabbat, ‘First day from Shabbat’, ‘Second
day from Shabbat’ etc.. This is part of the
Mitzvah that we are commanded to remember Shabbat every day. This is
the simple meaning of the verse and is also the explanation of Ibn Ezra.
It is unclear from this explanation if this is a halacha, per se, or if this is a statement of what Jews do by habit, or otherwise.
However, see This article which quotes the stringency of the Chassam Sofer, and claims that this was (is still?) the Hungarian custom.
According to Chasam Sofer (Toras Moshe, Parashas Bo), there is an obligation to begin one’s letters with the number of the day of the week (counting from Sunday), and the number of the month (counting from Nissan): “This is an explicit rebuke [to those who do not use the Jewish calendar]: We should write in our letters and similar documents the first day of the week, first month of the year, and so on, to testify that Hashem created the world in six days, ‘and He rested on the seventh,’ and to remember the redemption from Egypt—and not Heaven forbid, the date of the nations.”
I recommend reading the whole article as it also lists contemporary opinions. The ruling by Rav Ovadia Yosef permits using secular dates (he explains month names, not day names, though).