The advice seems to suggest if their options expire if you don't buy them at a certain point (except for leaving the company, which makes sense), say no and ask for cash. If they can't pay, now you know where you really stand.
If the liquidity event involves the public stock market in any way e.g. IPO, merger, acquisition with a nontrivial part of the proceeds paid in shares of a public company - then you are forced to execise on one hand, and have a lock-up period, usually 6 months, forced by the underwriters or SEC rule 144.
That is, there is a mandatory 6 months wait between the forced exercise and the effective liquidity event.
This applies to investors as well as employees, BTW: I was bitten by this as an investor -- a modest 5X return on investment after 2 years turned out to be less than 2.5X return (still nice), but the taxes were paid on the 5X numbers, and the end result net of taxes was therefore 1.1X -- and it would have been a loss if everything happened in the other half of the year (luckily, I was able to net the gains with the losses because the 6 month lockup was april-october; but had it been october-april, even that wouldn't have been possible).
In two other tax regimes I've operated in, you are only ever assessed taxes in the event you can take money into your pocket. The US system is ridiculously unfair in this sense.
I made the point that they didn't have to exercise when they did, and if they had waited they wouldn't have taken on any risk about the price of the stock they were receiving.
Note how those are not mutually exclusive.
The article refers to "stock grants" and refers to the fact that "Top sales employees were awarded with annual bonuses of 20,000 shares of common stock".
IANAL but AFAIK, receipt of stock grants would trigger an income tax liability on the value of the shares at the time they were granted.
Tax treatment of stock grants and options is very complicated and so dependent on details that it seems somewhat unfair to jump to any conclusions based on assumptions which are almost certainly incorrect.
There are very real tax advantages to exercising early.
They could have avoided this by waiting to exercise their options on the eve of the liquidity event. In this case there would have been no risk. But they exercised earlier presumably to start the clock on long term capital gains treatment for the stock they received when they exercised.
They took risk they didn't need to take and they got burned. It's worth keeping that in mind as another aspect to the story.