Southwest? You'd be better off at RyanAir or EZJet. Same type of operation with probably the same pay compensation---B-737s only, up and down 6x per day, short overnights and only $2500 USD to start monthly!
As far as American air carriers hiring foreign nationals, I can positively confirm it is true. There are many Irish, German, Fillipino, Indian, Canadian, French and others working for American and United.
Having flown VS and thoroughly enjoying the flights, I would gladly work for them if they'd hire Yanks. You must understand, since 9-11, there are over 8000+ furloughed/laid-off pilots from large US carriers looking for a minute number of jobs. I'm now one of them.