The basic connection in swing is far different than that the traditional ballroom position, both in the open and closed positions.
In both open and closed positions, be sure to keep a decent amount of "frame" or "tension". Some people will also refer to this as "giving weight". All this means is to be sure to give a small amount of pressure back to your partner - this will make it significantly easier to feel the moves and will help with momentum in turns.
The Open Position
In the open position, the lead begins with his arms bent at roughly a 90-degree angle at waist height with his hands curved and thumbs up (pretend you're holding a soda can, then point your thumbs up and you basically have the right shape). The follow's arms are similar to a position of pushing a shopping cart - similarly bent at about 90-degrees at waist height with palms parallel to the ground and the fingers curved (as to fit into the lead's hands). These are exaggerated descriptions, you can relax into more natural hand positions, but be sure to keep your arms bent and around waist level.
The Closed Position
This position is significantly closer to the traditional ballroom position. The lead puts his right hand on the follow's back approximately at the level of her shoulder blade. Keeping this hand high will help significantly when you go to lead a move that requires you to drop the hand to her waist. The left hand should be in a similar position as in the open position, approximately at the waist and bent. The follow should keep her left hand on the front of the lead's shoulder. Moving the hand to the lead's back will prevent you from feeling when the lead comes forward and can result in some awkward accidents and generally a weaker sense of lead. The right hand should be similar to the open position, approximately at waist height. A note about the left and right hands for the lead and follow, respectively - moving the hand up closer to shoulder height, in a more traditional ballroom style will prevent the lead from being able to effectively signal a turn out of the closed position. Plus, it's just more stylistically appropriate to keep the hand low.