GA pilot reported a near miss during departure climbout with aircraft entering traffic pattern at JCA airport.

Narrative

Following takeoff from Runway 35 at JCA as I was making my left crosswind turn at an approximate altitude of 1700 ft. MSL an aircraft which had not announced on the radio flew directly under us at close range during our departure. I had communicated our departure prior to takeoff and announced my crosswind turn. At no time did we hear the aircraft transmit their approach to the airport or their entry into the downwind for [Runway] 35. While I did have a Garmin GDS39 with ADS-B in the aircraft and setup on my tablet for situational awareness; for an unknown reason it was not receiving traffic data during our approaches and landings at JCA. During the crosswind turn and climbout my CFI spotted the aircraft on track to pass close by us from right to left and took rapid evasive action; expediting our climb and initiating a bank to the right. I did not have a visual on the other aircraft until after it had passed under us; at which point I was able to identify it and noted it was approximately 500 ft. below us and at its closest point I believe flew directly under us. It is possible the aircraft made a radio transmission after passing under us; however I did not note it's N-number. The aircraft was a low-wing; single engine piston aircraft. Other traffic in the area had heard our radio transmissions during our multiple landing approaches and takeoffs at JCA during this flight as we had communicated directly with other aircraft.Student (PF (Pilot Flying)) focused on aircraft control and looking inside the aircraft monitoring airspeed during the climb-out rather than scanning for traffic; difficulty of spotting rapidly moving crossing traffic on the right side of Aircraft X when obstructed by the right seat passenger/pilot. Instructor recognized the crossing traffic below us within 1 mile and reacted rapidly to take evasive action; but without positive transfer of controls; causing brief confusion over who was in active control of the aircraft while maneuvering. Lack of ADS-B In in the specific aircraft we were flying which failed to provide traffic warnings on the built-in G300 avionics system. Issue with personal Garmin GDS-39 not working which was identified on the ground; but rather than taking the time to attempt to troubleshoot and reset the device Student and Instructor opted to continue the practice flight; a better decision would have been to stay on the ground for the brief time required to reset the device and attempt to re-sync and ensure traffic was being received again. Tests post-flight show that only a power cycle was required to again receive traffic data. Failure of low-wing pilot to transmit his location on CTAF or be listening for departing traffic on CTAF. Failure of low-wing pilot to visually identify departing traffic on the runway he was on approach to.