No nukes will not solve the issue you would need a lot of nukes and the fallout would be harmful to us as well as them.
Aggressive border defense is the answer I think. With roaming squads of hunter killers supported by light armored vehicles and gunships. I can't imagine they'd be too eager to come here once they discover that they'll be turned into hamburger meat on sight by apache's and c-130 spooky's! Once the migration is stopped we hold the border and let nature take it's course in their own countries. Cut off all aid to them, all ties must be cut. Then we will see how much of a problem they are after 20 years or so.