To begin with we are faced with the issue of term
definition. How do we define
morality in the first place?
Let us for the sake of argument define it thus:
Morality is a complete set of ethical rules which is both imposed on the individual by an outside influence and total in its approach - one either lives by all of it all the time or not at all.
This is the more
religious definition of morality - religious laws allow for no argument or
transgression, are inclusive of almost all parts of our daily lives, and are
imposed upon us by a God or a religious establishment.
Although an argument can be made that the legal system of any given country answers these criteria, we will, again for the sake of argument, ignore this aspect of legislation (especially in view of the fact that so many of our basic laws draw their roots from earlier foundations of religious morality).
According to this definition it is clear that an atheist, or a nontheist, cannot have morals. However, a closer examination will point us to the concept of ethics. Now, while picking and choosing one's ethical principles from a set morality is, by definition, immoral, there's nothing to say that it is also somehow wrong.
A secular person can have a complete, coherent and workable set of ethics, some of which are borrowed from older moral codes and some of which are personal developments, to replace the old morality. Not only is this system not inferior to conformist religious morality, it is superior in that it allows for a greater measure of personal compatibility and intellectual development.
We can of course strive to teach other people our own ethics, and convince them that they are the best of any other system. However, unlike with morality, there is very little we can do to force our ethics on anyone else. So no, a secular person cannot have morals, but there's no reason to assume that they cannot have something even better. Lack of a religious regime does not equal personal ethical lawlessness.