Generally speaking, no, but that really depends on the severity of the depression or anxiety (or both) in a patient.

Some background: At the time Aktion Tiergartenstraße 4, the programme of euthanasia used on those considered 'handicapped' under the Nazis, was implemented, psychiatry was still limited, coming off of the reinvention of the practise in the late nineteenth century. Most of the patients killed under the euthanasia practises of both Aktion T4 and what came after it were described as severely mentally disabled and not actively contributing to society (which, under the practise of eugenics that was justification for these and all genocides during the Nazi regime, indicated they needed to be euthanised), which indicates that only those with severe depression and anxiety that manifested in a way that would disturb 'normal' life were murdered.

Psychiatric patients were chosen for assessment by psychiatrists on site at asylums, and if they were condemned they were transferred to the specialised asylums where they would be euthanised, so there was a very close relationship between seeking treatment for psychiatric disorders and being killed for seeking treatment. Relatives of patients who were murdered under Aktion T4 were given letters after the fact, giving false, made-up reasons for the deaths of their relatives, which contributed to the end of AT4 when these relatives became suspicious and involved the Catholic Church (which has a very complex role in the Nazi regime).

However, the killings didn't end after Hitler ended T4. They moved on to a second phase, which was less discriminating, as patients often did not have to be moved to a specialised asylum to be euthanised.

It comes down to the definition of psychiatric patients in the 30s and 40s, and what could cause someone to be committed to an asylum. Depression was certainly diagnosed and studied at the time, and there were multiple theories and categorisations (many put forth within Germany post- and including Freud), and it was usually recognised in a patient as more than mourning. There were cases patients admitted to asylums by their families or doctors during major depressive episodes.

Anxiety is less clearly defined and studied during this time period, often categorised as a subset of other mental illnesses, but again, it comes down to the severity of the disorders and whether or not a patient was considered useless in society.

Source (if I need more let me know, this is my primary):

Thorsten Noack M.D. & Heiner Fangerau M.D. (2007) Eugenics, Euthanasia, and Aftermath, International Journal of Mental Health, 36:1, 112-124

It depends. How severe is their condition? Have they received treatment? Is their condition chronic? It's also difficult to tell because some of the language used back then were vague and may or may not indicate what we would understand to be depression or anxiety. Among those referred for sterilization, for example, were people who were "work shy," which could indicate someone too depressed to work. But it could also indicate any number of other illnesses as well. Some people were referred for sterilization because they'd had "a nervous breakdown," something which today is not considered a clinical term for any particular illness. If someone was currently or had ever been institutionalized, it was more likely they'd be referred for sterilization.

In order for the regime to order sterilization, a doctor had to refer a patient to a court which tried people for their fitness to breed. The court had to take the evidence into consideration and make a ruling. The patient was required to report for sterilization.

As far as I can tell, veterans who were "shell shocked" were an exception to the sterilization court.

Sources - The Third Reich: The Rise, The Nazi Doctors