Can my green card be rejected on the grounds of common law marriage?

Generally, the USCIS will recognize a marriage if it was valid in the place where it was consummated. Few states continue to recognize common law marriage; some have important residency requirements; and some do only for marriages that were created before a specified date (i.e. before the state discontinued allowing common law marriages). Beyond that, proving to the USCIS the existence of a legally valid common law marriage can, itself, be challenging (beyond the usual requirement for all couples that they prove that they are living together in a bona fide marriage). Especially under the circumstances you described, it would be wise to engage an immigration attorney to represent you in a marriage-based immigration application.