Usually it isn't the store pushing this, but Square itself. They take a percentage of each transaction so they naturally want to make the charges as high as possible.
Yeah, we make POS systems and were asked for tipping functionality on the card machines as soon as we started selling to restaurants.
Recently we also added it so you can record cash tips as well, rather than them just being pocketed. Optimistically, this is so it can be shared more equally between staff rather than just giving it to the pretty girl who takes your money. Realistically, the owner is probably taking the lion's share.
As someone who uses square at our business, you can indeed customize it a little bit. Though I've never seen the "talk to a cashier button." Ours says custom tip where someone can place a zero $$ tip as well.
But to the point of square taking a cut... The entire amount (including tip) has a percentage taken by square... So they do want you to have as big a transaction as possible because they make more money.