Yes, they definitely can. A broody is a hen that wants to sit on eggs and hatch chicks. It is a hormonal thing, and can be difficult to break if you do not intend to give her eggs to sit on. If they have eggs to sit on, they sit, then mother the chicks until their bodies tell them it's time to go lay eggs again. That length of times varies, dependent on the bird (broody), and I would guess the time of year that they are brooding. This seems to be the case with our own broody. The length of time that she has brooded after hatch varies from about 8 weeks, to 14 weeks, seemingly factoring in the time of year she is brooding in---perhaps the amount of daylight hours. I have read posts here where there are pictures of a broody sitting on chicks that are older than 14 weeks, though.
And to answer the first question more clearly----yes, they certainly can be broody without a rooster. We haven't had a rooster for most of the time we have had chickens, yet our Cuckoo Marans hen, Tuney goes broody quite often. The urge to brood is so strong that they may sit on infertile eggs, golf balls, rocks or nothing at all.