In at least the early part of the Gemini program, the ground-side network still wasn't able to relay all ground stations to a central control center, so multiple sites still needed CAPCOMs, and not all of those CAPCOMs were astronaut-trained. In addition, the Gemini program featured longer-duration flights than those of Mercury, beginning with the second crewed flight, the 4-day Gemini 4 mission, so there was a need for multiple shifts worth of CAPCOMs.
During the run-up to Gemini 3, there was some conflict at Carnarvon between the local flight control team and astronaut Pete Conrad, who was assigned as Carnarvon CAPCOM. Possibly as a consequence of that incident, on Gemini 4, non-astronaut flight director Ed Fendell served as Carnarvon CAPCOM, while astronaut Dave Scott was onsite as an observer! The transcript confirms that Carnarvon CAPCOM communicated directly with the spacecraft.
Again on Gemini 5, Carnarvon had a non-astronaut CAPCOM, Chuck Lewis, directly in contact with the spacecraft.
I've so far been unable to confirm that there was a non-astronaut CAPCOM on the ground during Gemini 3 -- the first crewed Gemini flight. Wikipedia and spacerace.com agree that Gordon Cooper was the Cape CAPCOM and Roger Chaffee was the Houston CAPCOM; honeysucklecreek.net places Pete Conrad at Carnarvon and Neil Armstrong at Hawaii.
The Gemini 3 transcript records the following CAPCOM call signs:
- From liftoff: Cape CAPCOM (Cooper; Grissom addresses him as "Gordo".)
- From 15:16: Canary CAPCOM - unknown individual
- From 25:07, over the Kano, Nigeria station, but talking to Cape CAPCOM -- the ground-side network is relaying.
- From 35:08, over Tananarive station, Cape CAPCOM again.
- From 42:27, USNS Coastal Sentry (CSQ CAPCOM) - unknown individual
- From 50:27, Carnarvon CAPCOM (Conrad)
- 1:11:32, via the Canton ground station, Cape CAPCOM again.
- 1:26:31, USNS Rose Knot (RKV CAPCOM) - unknown individual
- 1:29:43, Guaymas CAPCOM - unknown individual
- 1:31:10, Texas CAPCOM (Chaffee)
From here we work our way back to the Cape. Late in the second orbit, we get one more station in the mix:
- 2:49:48, Hawaii CAPCOM (Armstrong)
So my unknowns here are Canary, CSQ, RKV, and Guaymas. Since a number of different sources mention the CAPCOMs at these sites without naming them, I believe they were not astronauts, but I can't prove it.