On an ADS-B In display, do the pilots see the squawk code or call sign for other aircraft?
ADS-B is part of a system that compromises a number of different things. ADS-B (In) means that your aircraft is capable of receiving the ADS-B data, FIS-B and TIS-B as well. FIS-B broadcasts information such as weather. TIS-B is a traffic broadcast system.

ADS-B includes a number of messages (about 27 different messages), including:


Aircraft Identification
Surface Position
Airborne Position (Barometric)
Airborne Velocity
Airborne Position (GNSS)
Aircraft Status
Target State/Status
Aircraft Operation Status


Each message packet holds 56-bits of data. The aircraft identification message includes the aircraft type, category, and callsign. 

Now, to a pilot, here is basically what you would see:


Source: NexAirAvionics.com

You can see that some targets include the flight number (JBU1075) and clicking on a target (this is in Garmin Pilot) will give a little more information on the target:


Source: Garmin.com

What the pilots don't see is the squawk code, this information is not useful to pilots because they don't really care about that for other aircraft. The only real use of ADS-B target information is to understand where other aircraft are. If a pilot needed to call up another pilot, they would do so by tail number or flight number. ATC Controllers and other pilots will never call a pilot by squawk code. 

Why? Because the squawk code changes potentially many times during a flight. Let's take for example a simple flight from KGRB to KMNM (a flight I'm familiar with):

On the ground at KGRB, when telling ATC about my destination they will assign me a squawk code, for example 2044. After departing they will call me up "1205C turn on course, altitude your discretion" and I will acknowledge. About 30 miles out (if I didn't get flight following) they will say "1205C squawk VFR, frequency change approved". Now my squawk code changes to 1200. I might then pick up Minneapolis Center if I want to change my destination and they will assign me a different squawk code. 

The one thing that remains constant there is my tail number. I know to listen for (and have mentally trained a "listen up" trigger when I hear) "1205-charlie" or "05-charlie" on the radio. ATC has been talking to specific aircraft using the tail number/flight number since the ATC system was born. Squawk codes match aircraft with targets on the radar (because before ADS-B all that was transmitted was the squawk code). This information is only useful to ATC and not other pilots.

Now if I'm cruising around and I see a target on my ADS-B that is headed in the same general direction I am, and I'm not talking to ATC, I can try calling that target using the tail number/flight number directly (this is done on the air-to-air band or a local frequency, don't broadcast on 121.5 or you'll hear "guaaaaarrrrrrd" from some obnoxious pilot). 

Either way ADS-B is useful for traffic avoidance. I don't really need to know where that airplane is going, why it's going there, etc. I just need to know what it is (big jets can cause wake turbulence issues for smaller aircraft), which way it's heading, and how fast it's going there.