Short answer: No, the number is not related to the total tickets United has available on that flight.
Longer answer: All airline tickets are divided into fare classes on each flight (fare classes have nothing to do with 1st class, business coach or whatever other than certain fare classes will always be in specific cabins). So whatever fare class they are showing has 6 tickets remaining. Normally, the airline sells those tickets starting with the lowest priced ones first, but not all of the lowest fare class tickets are released. As mentioned in the previous post, this doesn't mean you won't see lower prices later, or even more at the same price. It just depends on how fast the plane fills up and how much the airline wants to fill it up if you will see lower prices later. If the flight is not selling as anticipated, more of the lower fare class buckets will open. If it is selling faster than anticipated, those 6 tickets will disappear and be replaced by higher priced ones even though no one has bought them. United used to show how many tickets were available in each fare class for each flight on the web site so you could have some idea about how soon you needed to purchase your tickets to get a low price. That went away after the merger with Continental. It is all controlled by the yield forecasting software each airline uses and makes absolutely no sense to most people who don't work for airlines (and I'm sure most of them have no clue either!).
And as to the financial health of United: They are no worse off right now than any other US based airline and most other international airlines as well. Airlines lay off thousands as the wind changes direction. After 9-11, during the 2008 financial meltdown, and other difficult times many airline employees found themselves without jobs. Many were brought back as the travel needs of the public increased. It will be the same now.