The Greeks were all over the place.
Now, I agree with Daniel being a fairly late book, but a few borrowed words dont mean much. The prophesies are the sticking point.
The Case for a Sixth Century Dating of Daniel. Appendix 3 of Daniel: Faithful Discipleship in a Foreign Land Questions of Language
The late date of Daniel was supported by scholars who claimed that the Aramaic sections in Daniel belonged to a later period. However, more recent studies have found that the Aramaic used in Daniel was used in the courts and chancelleries from the seventh century BC on, and tends to support an early date for Daniel.[313]
The presence of Persian and Greek loanwords in the text of Daniel, primarily in the words for musical instruments, was long taken as a proof that Daniel was written in the Greek period after Alexander’s conquests. It is now generally recognized that there were many earlier contacts with the Greeks and Persians, including Greek colonies in Egypt in the mid-seventh century BC and Greek mercenary troops in the Battle of Carchemish in 605 BC. Also, the names of musical instruments could well be found along with the instruments at the Persian court.[314] Today, linguistic arguments for a late date of Daniel are considered quite weak.