Generally no. You would be unlikely to pronounce or name the punctuation symbols unless you were dictating to someone who was typing it out.
The news anchor may say quote/unquote to make clear where a quote begins or ends, but they are less likely to say "comma" when there's a comma. That would only be done if it was particularly important for the listener to have that information - for example, when discussing a legal document where the comma made a crucial difference to the meaning. Similarly, they are unlikely to speak out the brackets.
In the UK, if we did need to speak out the brackets, we would most likely use "bracket" or "open bracket" to refer to an opening parenthesis (, "close braket" to refer to a closing parenthesis ), and "square bracket" to refer to [, "end square bracket" for ].
Another practice sometimes followed is to omit the closing marker where it's obvious. For example, if two people were discussing the spelling "person(s)", they might call it "person bracket s". (They could also call it "person bracket s bracket", and would be less likely to call it "person bracket s end bracket" unless there was a need to be precise.)
Note that square brackets normally contain editorial notes or editorial amendments (sometimes including the insertion of missing words needed to make sense of an incomplete sentence). So your example below is odd. Square brackets are not simply an "inner" form of bracket. If "I think!" is a remark by the original speaker, we'd offset it by a comma or a dash, not by brackets.
She told me, "I don’t know, Jack. I've never been there before (and neither have you [I think!])."
For this sentence:
He said, "I know that my friend Stephen —the poor dear!— has found living on his own very difficult."
We would normally just use pauses and intonation, but if necessary we could say "dash" to indicate the dashes.
She told me, "I don’t know, Jack. I've never been there before (and neither have you [I think!])."
Normally, we would just pause slightly before the brackets and possibly use a quieter voice for the bracketed bit (if reading aloud to child, for example). We would only name the brackets if it was important to be precise about what we were reading. (Americans normally refer to "(" ")" as parentheses and "[" "]" as brackets, while in the UK we usually refer to "(" ")" as brackets and "[" "]" as square brackets. But in specialist fields, different terminology might be used.)
