I would probably do it like this, in contexts such as yours:
1. Say "You have to go to the meeting", not "You need to go to the meeting".
i.e. without the full stop (or comma, or colon, or semi-colon, as the case may be) in the first quote.
But exclamation and question marks may be included, e.g.
2. Say "You have to go to the meeting!", not "You need to go to the meeting!".
3. Say "You have to go to the meeting?", not "You need to go to the meeting?".
In the case of ordinary dialogue in printed matter, by the way, most publishers prefer to put punctuation marks within the preceding inverted commas, e.g.
4. "Yes, when I am upstairs," he answered; "but I can't walk upstairs now."
MrP