The 74HC595 can drive +/- 25mA on an output, which should be fine for driving a single LED. The single 4-pin RGB varieties on eBay etc. pull about 20mA.
So if you are using your 595s in a row/column time-multiplexed scan matrix kind of thing where 1 LED at the intersection is on at a time, it should work.
If you are driving an entire row/column of leds through a pin or all at once or something, you will need one of those more expensive, higher-output chips.