My husband and I have been here for two years under J1, respectively J2 visas. Our visas will expire next year. While here, our son was born, so he is an American citizen.
Is there any possibility that we can stay here legally under a different visa (or applying for green card), after J1 and J2 expire, based on the reason that our son is an American citizen?

Sorry but (NO) get an immigration Attorney asap before your visa expire, otherwise you will have to go to your country to apply for it.

it sucks!!!

No... your US citizen child can apply immigrant visa for parents when he/she reaches 21 years old, plus he/she must meet other requirements.

You can either maintain non-immigrant visa or return to your home country.