It depends. But, in general, no.
Startup employees generally get equity in the form of stock options. In order for employees to get preferential tax treatment on these options they have to exercise these options at the time they are granted (which can require a large outlay of cash) and then hold them for at least one year.
Due to the cost and risk of this exercise, most employees do not do this. Instead they wait to see if the company becomes more valuable (and liquid) and only exercise the options if so. At that point any realized gains are taxed as ordinary wage income.
There are a great many resources online on this topic if you are interested in more details.