"Flush" here is used when something is at exactly the same level as a flat surface.
For example, if you take two tables of the same height, and push them together, their surfaces will be "flush".
Or, if something is embedded inside a surface, so that it sits at the same level as the surface. The slot on some optical drives is a good example of a slot that is "flush" [1].
So on an ATM a "flush" slot would be one that is just a plain slot in the flat plastic surface of the ATM, without any anti-skimming device which sticks out the front.
Like the other people are saying, it means" flat" or "broken outer surface".
Are you familiar with poker? We have another use of the word `flush` in that context, https://en.wikipedia.org/wiki/List_of_poker_hands#Flush where all of the cards are the same suit. So it's a similar use of flush in this instance, that everything is the same. It's similar to this thread's use of "flush".
We should not forget that flush also refers to the time when a person's face becomes red.
As in, your face will go flush if you realize your credit card reader is not flush because you might have just flushed all your winnings from having a poker flush down the toilet.
Please keep comments constructive. This does not help the original questioner, who may not understand why it isn't a fantastic analogy. Please either refrain from the negative commentary, or enhance the conversation by clarifying why you feel it is lacking.
In this case the poker analogy is terrible because in this case flush is talking about a physical surface being unbroken while poker is talking about a collection of items all being the same.
"Flush" in the poker context has more to do with definitions 1 (Abundantly full. In later use chiefly of a stream, etc.: Full to overflowing, swollen, in flood.) and 3 (Plentifully supplied (esp. with money).) rather than 5 (Even, level, in the same plane with / Even or level with the adjacent surface.).
To be "flush with money" is to have a lot of money---in poker, to be flush with hearts is to have 5 hearts.
I know others have already answered, but I found this learner's diction version of the Merriam-Webster dictionary (and it looks a lot better than their normal site anyway).
Scroll down to the 4th listing, which is the adjective version. "even or level with another surface"
So the original poster meant that he thought the card reader should just be completely even or level with the front of the machine. I guess that would just be a plain slot.
Looking it up, you might mean it glows in some color?