I was hit with 31 reasons why the apocrypha are not Scripture, and found problems with all 31 reasons. I will not go into each on in detail (you're welcome) but there are some general principles. First off, the whole issue of canonicity brings up the whole question of authority (since all differences between Catholics and Protestants is ultimately a matter of authority). Science fiction author John C Wright has written a few articles on this that are far better than anything I could write. A couple of quotes (with links to the articles):
"...one cannot argue that the books of the Bible are canonical and argue at the same time that neither the Church nor any one has the authority to canonize them." - John C Wright
"The redacted sections of the written message provide a logically insurmountable paradox for the Protestant messengers. If their sole authority for the authenticity of their message is the written part of the message, then they have no authority to redact or remove parts of the message on any grounds. The cannot throw away the Book of the Maccabeus or Tobit or the Letters of James or anything else because they claim that neither they nor anyone has the authority to define the cannon." - John C Wright
Now, onto the 31 reasons... I was able to fold all 31 down to five different reasons. You get 31 reasons by repeating the same thing from a slightly different angle or by claiming each church father as a separate authority. Here are my five reasons. If you can think of something that is claimed by Protestants that is not in these five categories, please let me know - I'd by happy to add a sixth.
1. The Catholic Church rejected the books, then changed the canon in the 1500s. This is simply not true, as you can find the documents of the earlier councils in the 300s which list the same canon (for instance http://taylormarshall.com/2008/08/decree-of-council-of-rome-ad-382-on.html).
2. The Jews rejected the books - although they did not until at least 100 AD, which means they were accepted during Jesus' life time (yes, there were certain sects that rejected various sets of books, but of course that is true all throughout history). Additionally the Jews rejected the books of the New Testament, so if we are to accept the Jews as authorities for the canon we have to discard the entire New Testament. I've heard the counter argument that the Jews have authority over the books that don't include Jesus, but claim is ridiculous on the face of it as all of Scripture is about Jesus, as there is one God.
3. The Church Fathers rejected the books - but of the dozens of Church Fathers, there are only a tiny minority that didn't, and it is unclear if they themselves didn't accept them or they were saying that others didn't accept them. For instance, St. Jerome wrote that "I wasn't relating my own personal views, but rather the remarks that they [the Jews] are wont to make against us." (http://www.catholiceducation.org/en/religion-and-philosophy/apologetics/5-myths-about-7-books.html)
4. Jesus and the Apostles rejected the books - but they didn't. In fact Jesus refers to Maccabees as being a Messianic prophetic book (John 10:22-36). He also paraphrases Sirach in Mathew 7:17-20. There are other examples, but these are sufficient to dispute the claim. Nowhere do Jesus and the Apostles define or refute any canon of Scripture.
5. The books are "different" in some way (either in historical errors, discrepancies, doctrine, genre, etc.) and therefore should be rejected - but each of these arguments are arguments against many of the books of the Old Testament (and New Testament). These books are no "different" than any other book in the Bible is from any other. I won't go into specifics, as there are literally dozens of things that can be brought up, but suffice it to say that every difference you can point out in a deuterocanonical book can be found in a non-disputed book in the canon.
Should we reject all the books that are "different" in some way from some other book? In addition, there are many non-canonical books that are "similar" to canonical books - should we add them? Who decides what is "similar enough" to warrant inclusion? Again we have that issue of authority
Ultimately the historical fact is that the Pharisees removed those books, along with the entire New Testament, after Jesus' death, in order to suppress Christianity. Christians accepted those books, with few exceptions. The Christians held several councils to resolve the matter and the issue of the canon was settled in the 300s All Christians accepted them as canonical for 1200 years until Martin Luther unilaterally changed the canon. His authority was not enough to remove all the books he wanted removed (such as James, Jude, Hebrews and Revelation) but citing the Jews as authoritative he was able to make those 7 "stick."
Luther made no bones about why he wanted to change the canon - it didn't fit his theology. In fact, in the Old Testament he also wanted to jettison the Pentateuch, Job, Ecclesiastes, Esther, Jonah (he couldn't find an excuse to do so, but his commentary on these books says that he didn't consider them to be canonical). He even added the word "alone" to Romans 3:28 to make it say "For we hold that a man is justified by faith alone apart from works of law." in order to make it support his theology.
So, it seems that the real reason for dropping the apocrypha from the canon of Scripture was to change God's word to fit Luthers' word, and that is just plain wrong.