The duration of the excursion is not clear but it doesn't seem problematic to be away for a long weekend - do they themselves work the office and actively contact people to offer jobs over the long weekend? Clearly, to the letter, advise them of the absence and be available to return.
I think the who's-paying issue is more relevant to welfare and disability support programs awarded on the basis of financial need. For those programs, benefits may be clawed back if other financial resources are available to a claimant. However EI is an insurance program. Benefits are due if premiums were paid in and eligibility requirements (unemployment) are met.