In the Oration we find something similar: "Security to the persons and properties of the governed is so obviously the design and end of civil government, that to attempt a logical proof of it would be like burning tapers at noonday, to assist the sun in enlightening the world;..."
This is a similarity in that it states that certain rights are self evident.
The oration goes on to say: "and it cannot be either virtuous or honorable to attempt to support a government of which this is not the great and principal basis; and it is to the last degree vicious and infamous to attempt to support a government which manifestly tends to render the persons and properties of the governed insecure."
Whereas the Declaration follows with: "That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn that mankind are more disposed to suffer, while evils are sufferable than to right themselves by abolishing the forms to which they are accustomed."
Again a similarity in that they both encourage resistance to a government that does not protect the rights of the people.
Another similarity is that they both accuse the British government of being tyrannical in its treatment of the colonies.
Both accuse the King of refusing to acknowledge the colonies' right to self government.
Both mention soldiers murdering innocent people and then escaping justice(Almost certainly a reference to the Boston Massacre)
Both accuse the British monarch of stealing from the colonies.
Both appeal to a Divine Creator that fighting for their freedom is the right of free people.
I had not read the Boston Massacre Oration, though I am not surprised to find it has so much in common with the Declaration of Independence. Most of the Founding Fathers of the USA were products of the Enlightenment and thefore likely to hold similar points of view(especially given that both Hancock and John Adams (cousin to Samuel Adams, an early friend of Hancock's) who pushed for Thomas Jefferson to draft the Declaration of Independence were so supportive of Independence. The final copy of the Declaration was of course editted by Congress and the first signature was non other than John Hancock's.
I hope this is helpful. I am by no means an expert on American History though I have read quite a bit about the founding fathers.