Looking at the numbers and making some estimates...
$10 billion in transactions probably turns into about 1% in fees, so we'll say revenue is about $100 million. Being generous, let's say they are running on a 30% margin. That would put profits around $30 million a year.
That's pretty good, but when you look at that number it would be a 10 year payback period at $300 million. At $1 billion, that is a 30 year payback period.
Given the $69 million in outside funding, I'm sure they are pushing for a 10x return, so $690 is probably the ballpark of the asking price. They are probably asking for $1 billion in hopes of a counter around $750 million.
If I were a company looking to acquire a merchant services provider, I would look at the cost of acquiring that $10 billion a year in transaction volume. Is it less than $1 billion? Probably. That is probably why MasterCard built this: https://www.simplify.com/commerce/ instead of dropping $1 billion to buy stripe or Braintree.
If you didn't have a big strategic need, the right price for Braintree is probably more like $200-300 million. Braintree won't sell at that valuation because they have raised too much money and it wouldn't be a big enough return.
I could see PayPal buying Braintree, just for some of the more developer focused company culture, but I don't see why Square would buy them.
My guess is Square and Stripe end up merging at some point. It almost feels inevitable.