Credit card processing fees when connected to Square are as follows:
2.6% of the total amount charged, plus $0.10 when the card is physically processed through a Square card reader
3.5% of the total amount charged, plus $0.15 when the card information is manually entered, like when a client makes a purchase online or when a client's card is charged for their membership.
If you are viewing Square's reports, you may see the total fee charged for a manually entered transaction as two different fees: Square fee and Application fee. The Application Fee portion is because the charge was processed through another application. In this case, that application is MassageBook. However the total of both fees will amount to the 3.5% + $0.15 rate.
You can view a full list of Square's credit card processing fees, here.