Unfortunately, Square doesn't support cards on file or passing in card numbers which is required for API integrations with HomeAway, booking.com etc. They only support the model where you send a Square form or embed to the guest and they key in their card numbers that way. That's why we haven't gotten to adding support for Square at this point.
Lynnbrook has different pricing models depending on your volume and mix of cards, so they can get you the best price for your situation, which is why they don't have a published price that applies to everybody.
If you want a processor similar to Square, look at Stripe. They're also 2.9% + 0.30 like Square.
