Welcome to the forum Peter & Vivian,
You can find the Everest specs & info in the 2007 brochure, posted here:
http://keystonerv.org/forums/showthread.php?t=2133
Given the fact that the 344J is a larger/heavier 5th wheel, a 350/3500 diesel truck would be the minimum. We've never towed a 37 ft long trailer, but those members who do, can give you a better opinion as to whether or not a dually might the best match (for stability) with this trailer.
As for truck 'brand'... while towing, your fuel mileage will probably be fairly similar regardless, so I'll simply suggest that you test drive each 'brand' & choose what best suits your preferences, in addition to matching truck towing capabilities to the trailer.
There's good info in the posts suggested by Festus2.
Good luck with your decision & happy camping!
Terri, the Chevy co-pilot