I've had a friend who got caught because he went to the DR for a week while on EI. They told him they get the info off the customs forms one fills out upon returning, and that if he'd have driven across the border, they'd never find out
Otherwise, foreign travel does not make one not eligible - one can just click the 'out of country' box while doing their bi-weekly online reports, and simply not get paid that week, it has no affect on other weeks in the claim.
I dont see how you paying her expenses, and especially you being SE has any relevance to this situation?????