In my exit interview as well as my resignation meeting, I was being pushed and pressured to give the name of my new construction employer. I do not wish to disclose this information. What's the deal?

You are under no obligation to disclose this information if you do not desire. If you want to wait, simply inform your boss, when you are settled in your new position, you will gladly give your contact information. Typically, asking for this information is just being nosy. Sometimes, employers will use this as leverage and tell you everything that it wrong with your new employer. Also, HR personnel will like this information to gauge if they are loosing talent to a competitor. Basically, it is up to you and your right to keep tight lipped if you so desire.