Upload & Sell: On
Square (or any other credit-card processing capability) is useful as a way to facilitate payment. I also use Square for credit cards, and it's painless (except for the 2.75% transaction fee, which for big transactions sometimes seems worse than it probably is).
Square isn't directly related to your contract -- you still need to have a contract and a means to transmit and execute it -- but it does mean you don't have to do everything in person. And it can also mean clients who might be waiting for a paycheck might be willing to make the payment on credit. I don't like to encourage people to borrow, but I won't turn down payment if that's the way the client wishes to provide it.
You can do the payment over the phone, and assuming electronic agreements and signatures are valid and straightforward in your state (I have no reason to think they aren't valid in Washington), you can then email the agreement and have it signed either electronically or have the client print, sign, scan and email back.
The more simple ways you offer to execute the agreement and accept payment, the fewer obstacles get in your way.