Roosters are mature male chicken, which does not have capability to lay eggs. The hen refers to female chicken which can lay eggs. No roosters don’t lay eggs at any circumstance.
Just like humans, males can’t give birth to the baby. Male and female contributes to the genetics of the baby, but only a female can give birth.
As we all know hens lay eggs. But have you ever asked, whether a rooster is needed to produce eggs? The answer is not always a rooster is required to produce eggs.
The eggs hen produces without a rooster are unfertilized eggs. These eggs will not hatch. The eggs produced after mating with rooster are fertilized eggs. Under the right incubation conditions, the eggs bear chicks.
There is a terminology called a cock egg. An egg that does not contain yolk and it can’t hatch. Previously people use to believe that these eggs were laid by roosters or cocks.
The scientific evidence on cock egg or cock’s egg or dwarf egg or witch egg occurs mostly with pullet’s first effort. This happens when the reproductive system is not fully matured. Read more on the wiki
The cock egg can also happen sometimes in mature hen, when the small piece of reproductive tissue breaks away, initiates the egg-producing glands to treat it as the yolk and wrap it. Then this forms the egg without any yolk in it, but some sort of grayish tissue.
The cock eggs even happen with guineafowl and quail.