If there is no God, then there is no reason to fear death or punishment.
The myth that atheists have no reason to fear death or punishment is one of the oddest and most difficult to understand myths I've ever encountered but it is a real one which I've seen expressed by Christians. Not only is this myth basically the opposite of what reality is, but it doesn't even appear at first glance to contain an expected criticism like these myths usually do. So what if atheists don't fear death of punishment? Why is this a problem?
First, there is something very bizarre about accusing atheists of not fearing death or punishment when religious theists, and especially Christians I've seen proclaiming this myth, are themselves can be taught not to fear death and punishment. They are taught not to fear death generally because there is an afterlife and if they accept Jesus Christ into their hearts, then they are guaranteed eternity in heaven with Jesus and God.
When Christianity was still a minority religion in the Roman Empire, Christians welcomed the death of martyrdom because it ensured that they would definitely go to heaven. Muslims today are, in many places, taught that they can become martyrs by launching suicide attacks against the enemies of Islam. Religious theists like these not only don't fear death, they practically welcome it.
Religious theists are also often taught not to fear punishment, if it's a case of being punished for their beliefs. Thus if people are threatened with punishment for refusing to cease activity which their religion requires, they are supposed to not fear that punishment and accept it willingly because there is virtue in being willing to suffer persecution for your faith.
If atheists don't fear death or punishment, it's probably in similar circumstances: they are willing to risk punishment or even death if it's in the defense of important values and beliefs. There is, however, nothing about atheism itself which would necessarily lead to a person being especially impervious to fear of death and punishment.
If anything is the case, it may be that atheists could be a little more inclined to fear death than theists because atheists don't generally believe in an afterlife and so they know that this life is all they have just one chance to experience being alive and then that's it. This doesn't mean that all atheists definitely are more afraid of death than others, it's just a reason why it's possible. It's certainly a lot more likely than the nonsense that the absence of belief in gods means that there is no reason to fear death.
If this myth is so obviously false, why do some people proclaim and believe it? The best reason I can think of is that death and punishment are common consequences for immoral behavior and are thus important means used by authoritarians for maintaining social order. Conservative, evangelical Christianity is driven in many ways by a excessive demand for social order, obedience to authority, and adherence to tradition. This is in turn due to a negative reaction to a world in which traditions and authority cannot be taken for granted as they once were.
Atheists, by dismissing much that these Christians take for granted, may be perceived as a threat to social order unlike any other despite their relatively small numbers. Thus myths like this could be an expression of what people perceive the threat of atheism to be and why it's a threat. This is a poor reason for such a myth because there is no evidence that atheists are actually a threat to social order; what little data does exist suggests that atheists may be more law-abiding than theists. It is, however, the best reason I can come up with any other reasons are likely to be even worse.