Not by choice. I believe some regionals will fly under the banner of different majors, depending on the scheduling and the routing, but it would be determined ahead of time. If you are a SkyWest pilot then you are a SkyWest pilot, you can't just decide that you want to be a Horizon Air employee because you flipped a coin that morning.
I don't know exactly the standards at US airlines. When I worked the IT at a charter airline in Germany, they had a lot of freelance pilots on call to fill their cockpits, and those pilots flew for other charter airlines at the same time.
What VZ626 means, however, seems to be a totally different question. I think he/she/* means the regionals that serve more than one network. SkyWest, for example, flies for either United Express, Delta Connection and American Eagle with planes in the respective paint job (for example, they have 61 CRJ-700s in American Eagle livery, 5 CRJ-700s in Delta Connection livery and 19 CRJ-700s in United Express livery; they also have 32 E175s with Alaska Airlines livery, 34 E175s for Delta Connection and 65 E175s for United Express) with their own SkyWest callsign, while simultaneously having an "official" flight number (for the pax) with the respective AAL/ASA/DAL/UAL ICAO code. The same goes for Republic Airways who operate their planes for American Eagle, Delta Connection and United Express with their own "Brickyard" callsign.
I think the question was whether it's possible that a New York based pilot for Republic Airways drives to JFK on Wednesday morning to sit in the cockpit of a Republic E175S for Delta Connection to fly legs from JFK to JAX (N212JQ, RPA5752/DAL5752), from JAX to BOS (N212JQ, RPA5778/DAL5778), from BOS to BNA (N212JQ, RPA5629/DAL5629), stays the night, then flies a Republic E170 for United Express from BNA to EWR (N733YX, RBA3408/UAL3408) and EWR to ORD (N733YX, RPA3577/UAL3577) - and then probably deadheads back to LGA on a regular American Airlines flight on Thursday.
That I don't know. But since it's the same company, the same uniform and (I think) the same typerating for the E175S and E170, it should be possible. That could be just a matter of crew scheduling.