Can cross the border on a green card?

Yes provided you have applied for renewal and have the receipt handy. Else it would be a problem to reenter. Only under unavoidable circumstances and on humanitarian basis for a very important urgent trip this would be acceptable.

All eighteen years of age or older residents permanently in the United States must carry their valid physical card itself at all times. According to Immigration and Nationality Act section 264(e) failing to do so would result in penalty of a fine not exceeding 100 dollars and/or imprisoned not more than 30 days for each offense. The federal government can only impose these penalties.