Stock grants are possible, but you would owe regular income tax on the FMV regardless of whether the shares are actually liquid, or will most likely end up worthless.
Your shares (or options) dilute exactly as you would expect. The company can issue as many new shares as their Bylaws allow and their stockholders vote for, and your percentage goes down. More importantly, inquire about participation preferences of preferred shares, which will have a liquidation preference; you do not want participating preferred, only participation on a as-converted basis (these are the investors shares, not yours). That means in a sale investors either get their share of a liquidation preference, or the share price, but not both.
Also, your math ("just multiply their shares to reflect any new shares that are issued") doesn't work. Try it on paper and you'll see why :-)
What I'm really trying to understand, is how a person can go from owning a significant percentage of a company, to having his shares become worthless, even if the company is worth quite a bit. It seems like any system of regulations or laws that allows something like that to occur must be fundamentally flawed.
I've taken some economics classes, but they don't really delve into issues like this. Can anyone recommend a good resource for learning how this works?
If you work through some common scenarios, you'll see how the concept of owning a 'percentage' breaks down really quickly, and the concept of owning a 'share' actually keeps things very simple.
There's nothing about it that's for the purpose of fucking people out of money, but like many things in life, it's perfectly possible to achieve that effect.
Signing up for a straight RSU/stock deal is rarely advantageous for a private company employee - stock grants are issued (and taxed) at vesting time, while ISOs are taxed at exercise time.
You control the latter, but not the former, and the risks of getting stuck with not very liquid private stock and pending tax bill with stock grants outweigh the benefits of not having to pay income taxes on their appreciated value. Generally the only companies even offering stock/RSU grants are those with either high liquidity in private markets or with public offering in pending mode.
Employee equity is diluted with each new financing event, as typically new stock is issued to support the round. Sometimes investors would insist on preferred stock with liquidation preferences, which doesn't necessarily dilute the employees, but creates interesting situations if the company is to be sold.
One thing to keep in mind, is that most start-ups fail, and most successful start-ups don't turn the early employees into millionaires. I've often heard the advice that you should just look at any equity/options offered as a bonus, and negotiate for a fair wage. At the very least you should go in understanding that you are potentially giving up income for equity that may or may not ever be worth a lot of money.