This is not an immigration law question.
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Whether they can will not be determined by the 4 corners of the US immigration law. You are in a wrong forum.
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This appears to be an insurance question, not an immigration question. I will change the practice area.,
Although AVVO describes this site as providing free legal advice, it is really a simple Q&A forum. The volunteer attorneys provide general answers. No specific legal advice is given here and no attorney-client relationship is established. For precise direction and legal advice, please consult in person with an attorney in your area. Be sure to bring all relevant paperwork with you.
The simple answer is yes they can.
Your insurance policy likely requires you to answer questions and provide information necessary for a reasonable investigation. It is not uncommon for an attorney to determine during an examination under oath that there are additional documents that need to be reviewed that they have not yet been given.
I would suggest that you speak with an attorney about this sooner rather than later. The exam under oath has occurred because they suspect there may be a coverage issue in play - if there is an attorney can provide you with advice on how to protect your claim.