Is having a green card enough or do we have to be US citizens in order to receive social security and Medicare benefits after reaching age 62?

First, you have to pay into SS to get benefits. You need 40 quarters, 10 years, to qualify for a basic benefit. Typically Medicare starts at 65, not 62, unless you qualify under special conditions.

What ma0641 said -- andyou don't have to be a citizen to get SS retirement.