(1)If you work more than 1 yr and company is big enough, they have to give you family leave by law (Unpaid, upto 12 weeks).
“Employees are eligible for leave if they have worked for their employer at least 12 months, at least 1,250 hours over the past 12 months, and work at a location where the company employs 50 or more employees within 75 miles. Whether an employee has worked the minimum 1,250 hours of service is determined according to FLSA principles for determining compensable hours or work.”
If you are eligible, your company denies – you can file a compliant to labor department (and can go law suit if you wish).. But I believe notice them you are ready to do that, they will grant you leave.
(2) Green card: You already work more than 1 yr, and you had intention to work when you applied green card (not fraud), it is hard for them to take away your green card.I believe they just threatening with without ground. But review what contract says – Many company (even big US company) requires certain number of years of empolyment when they sponsored green card. When it violate they normally requires for you to pay back all the expense they spent green card process. This is perfectly legal and not un-common and unfair. So, find out what contract says when you violate contract.
(Sorry I cannot type in Korean. I’m in the business trip with company computer….)