No. We are German Citizens and we have bought and sold houses under J, H and Green Card, no difference. The one thing you need to be is legal, if you do not have a SNN get a Tax ID number. Best is to go to a local bank.
Only difference might be if you sell and you are moving back to where you came from, if there are any taxes due the title company will withhold them.
Call me if you need more information
541 207 5551