You may not be able to prove that you have the legal right to live and work in America, if you hold an expired Green Card. This resident card is not permanent and you need to apply for renewal if you find that your card is about to expire. However, the lawful status bestowed upon you may not expire even if your Green Card expires. Foreign nationals who are granted permanent resident cards can travel abroad with their Green Cards, if they wish to travel abroad for less than 180 days and they can obtain re-entry permits and travel abroad for more than 6 months. However, in both the cases, to travel abroad, a permanent resident must hold a valid Green Card.
If that card is invalid, you might get into trouble as Green Cards are the only identity documents that are issued to permanent residents as a proof of their lawful status in the country. If you travel abroad with an expired resident card, in certain circumstances, you may not be able to get back to the country. But you can get help from the nearest US Consulate or port of entry, if your Green Card expires while you are in a foreign country. Likewise, if your card is expiring and if you have planned to get back to America prior to the expiration of your Green Card, you will have to file Form I-90, to renew your expiring Green Card, as soon as you get into America.
In some cases, airlines may refuse to board permanent residents who hold expired Green Cards. However, a permanent resident may be permitted to enter into the United States, if he proves that he does not have intentions to remain in a foreign country permanently, but this will delay his travel. Hence, in order to avoid such difficulties, you may submit your Green Card renewal application, before leaving the country and you can do so only if your card will expire within the next six months. You can contact the US Consulate in that foreign country for information on renewing your Green Card, if your card expires while you are outside the United States as you cannot file Form I-90, while abroad.