ATC calling out regional airlines in traffic advisories: phonetic or "aircraft paint" airline used?
The call-sign is not used – it's as useful as saying the tail number if it's a GA plane ;)

The information given to the pilot is in FAA JO 7110.65 – section 2-1-21. TRAFFIC ADVISORIES.


Azimuth (o'clock position), or cardinal position (e.g., northeast) if moving rapidly
Distance
Direction and/or relative motion (e.g., southwest bound, converging)
If known, type and altitude.


Example:

ATC: [You], traffic, 12 o'clock, 10 miles, opposite direction, Boeing 737, one thousand above.



A comment by @user71659 remarked that Cockpit Display of Traffic Information (CDTI) can be used to follow traffic, and hence a call-sign/tail number would be given. This notice was cancelled in May of 2016 and was only to authorized users.