Your ex-husband cannot sue your boyfriend. He can take you to court to ask for an increase in child support. If you are unemployed, you should have taken your ex-husband back to court to DECREASE child support that you pay. You still can. Your money -- the money you yourself earn -- is the only money accessible to your ex as child support. If you earn nothing, he can access NOTHING. $600 a month for Prep School is discretionary, and if you don't care if your son goes to Prep School or not, the court should not make you pay for it -- even if you had an income.
Go to court now and get you support payment decreased. File the OSC before the end of the month so that the decrease will begin on the first of September (no matter when it's decided).
Slightly different answer (but substance the same) - the ex "can" sue the boyfriend, but won't get anywhere. Anyone can sue anyone basically, whether they get anywhere is a different story.
Sounds like you may be entitled to child support if you have 0 income. Also, the Judge will not be happy that your ex is denying you visitation - your husband is digging himself a big whole.
If he violated the court order, then you can bring a contempt action.
Good luck with your situation and remember the above is general information and not specific advice since I haven't seen your paperwork.