JetBlue vs United vs American?

I live and work in coral gables, florida and my company just told me they're going to ask me to fly up to northern New Jersey about once or twice a month for a meeting with another branch's employees. I'm getting a raise but the flights will technically be out of my own pocket. I'm considering which airline would be the best option to become a member with and fly with most or all of the time. JetBlue only flies to Newark from Fort Lauderdale which is further from my house. United and American both fly from Miami but they're both pretty bad Airlines, so what balances it out would be the loyalty programs. What should I do?

I'm not sure what you mean by "bad airlines". They are both major US airlines, with excellent safety records. United has had a few unfortunate incidents in the news, but that does not reflect most people's experiences. All US airlines squeeze in the seats for maximum occupancy and minimal comfort, and none serve you food anymore. However, United and American fly everywhere- so in terms of building up miles to use for other travel, they would be much better choices than Jetblue. However, if there's a major difference in the fares, the value of the miles probably wouldn't make up for the difference.
I've never heard of an employee paying for his/her own company travel.

Why would you take a job where you are paying for company travel out of your own dime?
- seems like a bad company to work for… Are they treating you as a contractor when you aren't?

There are virtually no differences between flight point accumulations. 1 airline point for each flight mile (unless you are purchasing discount fares). "Cheap" point redemption is about 25,000 points for a domestic flight. So you will earn about 1 free domestic flight a year.
- Jetblue has limited destinations, so I'd probably stay away from that.

What makes a bad airline to you? A company that someone decides to take a video of an event and then post it online?
- I have flown about 1,500,000 miles on American. I have never experienced anything like those posted online. I would imagine that those flying United will also have no experience with those few bad seeds.

The company is turning a legitimate business expense into a personal expense that is difficult to write off taxes. They are totally mismanaged in accounting. You'll save much more if they subtract the air ticket from gross pay - pay you less by the cost of travel if that's their game with you, because IRS rules on unreimbursed business travel are terrible.
JetBlue has more comfortable seating, free premium unlimited snacks, and better on-board entertainment.
For northern NJ, probably flying into Newark is better for travel to avoid the city.
The frequent flyer program is the least concern.
United has 5 non-stop flights a day into Newark from Miami
Frontier only goes to LaGuardia, Delta has no non-stops to Newark from Miami
American has only 3 non-stop to Newark
You are correct that Jetblue is only out of Ft Lauderdale.
Your standard airline is United.

I used to travel a lot - for several different companies. Never, ever, had a situation where I had to pay for travel from my own funds and not allowed to submit an expense report for reimbursement.

That sounds fishy to me.

Any flight, on any airline, can either be wonderful or terrible. I have had good and bad flights on many airlines. I have to say, though, that the vast majority of all flights have been OK, regardless of the airline. If you were to ask every passenger deplaning from a flight, some will say it was perfect, some will say it was horrid - but they were all on the same plane at the same time!

Choose based on the convenience for you. If I was going to NYC, I would rather not fly into Newark.