Is it legitimate to ban the burqa?

The state has absolutely no business involving itself in matters of attire. Obviously it can say that a woman's face should be visible when, for example, she is driving, using social services, or in court; but if a woman wants to go about her business fully covered she shouldn't have to explain herself to anyone. However, if a shop-owner insists that all customers' faces are visible when they are in his premises, the state should have no right to butt in and dictate his 'policy' either.

Should an employee be allowed to wear a cross at work?

Of course....unless his employer doesn't want his staff to wear one. If an employer has the right to ask employees to cover tattoos, remove facial piercings, be clean-shaven, or wear a shirt and tie, he should be able to insist that staff remove any piece of jewellery, religious or purely cosmetic.