I sometimes see square brackets used while quoting. My assumption is that they are replacing a pronoun with what the object of the pronoun, but I never know for sure because I don't usually get to see what the original quote looks like before the modification. What are these called and what are the rules of use?
These are used to indicate that a direct quote has been edited — to fit the surrounding information, or to add context that does not show up within the scope of the quote. This page has a more detailed description:
They are simply used to add contextual clarity where the meaning is unclear. This is not only in quotes, but that's its most common usage:
Square brackets are not ONLY used in quotes. They are used often in translation. For example, the bible. Although not all bibles do this, one notable bible that did do this was "The Geneva Bible"
The Geneva Bible translators gave particular attention to retaining the flavor and sense of the original Hebrew. Words that the translators considered to be necessary additions were shown in italics, and text that had been added for grammatical clarity appeared in square brackets.
Example: Daniel 1:7
Square brackets, whether part of a quote or just in text simply mean: "Added for contextual accuracy"