No, you will not get free medical insurance if you get greencards.
Your kids need to sign I-864 affidavits of support ... guaranteeing that THEY WILL PAY for your needs for the first 10 years that you are in the US, or until you become US Citizens, which takes about 5 years.
Sorry the answer is no ... the spouse of a greencard holder can not stay in the US while they are sitting on the 2+ year waiting list.
I suggest he go back to school for the next few years ... or find a cap-exempt H employer.
I do not think that this is the 'final' word in this area. Things are warming up for the US Supreme Court to step in.
For people who don't know what you are talking about, here is a link to the case:
Since your OPT expired ... stop working ... immediately.
Talk to the lawyer that did the H papers to see if there is an RFE or appeal possible.
You currently have no status and ... depending on how the denial is written, you may be accruing unlawful status.
Meet with the company's immigration lawyer ... not someone in HR .. not an immigration 'team' ... a real live lawyer.
My colleague is correct ... 90 days prior to 1/15/2015 ... sometime in mid-October 2014.
WARNING ... don't file early ... they will reject your papers.
Also, save ALL bills, bank statements, etc. with BOTH of your names on it ... starting NOW. The biggest mistake people make is to only send a couple of 'recent' joint bills, which in your case would be only Sept/Oct. 2014.