In publicly traded companies the shares are taxed when they vest. So when they give you the 100k in stock, you are taxed on the 100k (usually they withhold a portion of the stock in vesting programs).
When you exercise them, the company's most recent valuation will include a value for the common stock you purchased, and you can get hit by AMT on the value increase from the strike price to the current value.
Note: I am not a tax expert and my memory is imperfect.
You can get in real trouble with options of non-public companies and with ESPP plans. Stock from an ESPP plan, if sold in the first 12 or 18 months of holding (depending on the plan details) will be reported as ordinary income at the time of purchase. If you sell during that 12-18 month period at a loss you will still owe taxes at the full price, and that can get you in a hole. I've heard of people owing 20k taxes on a 5k return because of a panic sell during a market downturn.
I'm just an engineer and not a tax professional, so there may be errors in what I've stated.
... you will never go underwater from holding onto a vested stock grant. (This is the situation the OP was asking about.)
... you will never go underwater from holding onto a vested & exercised stock option if you paid taxes out of pocket.
... you may go underwater if you take on debt to finance the tax liability incurred when exercising stock options and the value of the retained stock subsequently drops below the unpaid portion of the debt you undertook to exercise it. (This is the scenario that led to the stories you refer to.)
Now you DO have to worry about the taxes on capital gains. I recently had to deal with this because I sold my RSUs on the day they vested, thought the 3rd that auto-sold covered me, but didn't realize the .88 cents that the stock rose in the hour before I sold the stock qualified as cap gains and I needed to deal with declaring it. Short term cap gains in this situation basically is just your standard tax rate.
If you don't need the money and you think the company's stock is going to do decently over the next year then holding on to your shares for a year to get to long term cap gains isn't necessarily a bad idea. Depends on the company.