Yes, anxiety can make the muscles in your esophagus spasm and temporarily stop working . Anxiety can do even worse things such as stimulate spasms of the coronary arteries that can cause angina pectoris or even infarct. The power and strange abilities of human mind are unbelievable.
There is a group of psychiatric disorders called "somatoform disorders". Somatoform disorders convert psycho-emotional suffering into a symptom of an organic disorder, although there are no organic changes present in that organ. In fact, the patient is not ill but he thinks that is ill (hypochondria).
There is another group of diseases called "psycho-somatic diseases" wherein long lasting psycho-emotional suffering causes functional disorder first, followed by an organic disorder later. Such diseases are hypertension, peptic ulcer, tireotoxicosis, irritable bowel syndrome, colitis ulcerative, psychogenic asthma ....and others. The coommon thread among them is that psychological factors are important in their etiology.