You would like to ask the question to hospitals.
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. Consult with a qualified attorney before making any legal decisions. Gen Kimura, (832) 247-6932.
Getting residencies is very competitive. Unless the application specifically asks for your status, I would not bring it up other than to say you are authorized to work in the U.S. and not limited to any particular employer. [Why give them reasons to potentially rule you out due to concern that your status may not be extended?]
This general information does not establish any attorney-client relationship. There may well be factors not mentioned in the question which could and should be addressed in an attorney consultation.