I know there are "positive" and "negative" symptoms of schizophrenia as well as cognitive issues, but can a patient with schizophrenia have only the negative symptoms? If so, can there also be a patient with only positive symptoms?

Not likely….. the negative symptoms of Schizophrenia are just like the symptoms of Depression…. iof you have only negative symptoms you most likely have depression. The positive symptoms also can occur in different disorders like Bipolar….. if they have only positive symptoms they will nto diagnose Schizophrenia but try to decide which disorder is causing the psychosis as noted below it has to be determined that psychosis is not due to Bipolar……

Here is the DMS criteria for diagnosing Schizophrenia:

Characteristic symptoms: Two (or more) of the following, each present for a significant portion of time during a 1-month period (or less if successfully treated):

delusions
hallucinations
disorganized speech (e.g., frequent derailment or incoherence)
grossly disorganized or catatonic behavior
negative symptoms, i.e., affective flattening, alogia, or avolition

Note: only one Criterion a symptom is required if delusions are bizarre or hallucinations consist of a voice keeping up a running commentary on the person’s behavior or thoughts, or two or more voices conversing with each other.

Social/occupational dysfunction: For a significant portion of the time since the onset of the disturbance, one or more major areas of functioning such as work, interpersonal relations, or self-care are markedly below the level achieved prior to the onset (or when the onset is in childhood or adolescence, failure to achieve expected level of interpersonal, academic, or occupational achievement).

Duration: Continuous signs of the disturbance persist for at least 6 months. this 6-month period must include at least 1 month of symptoms (or less if successfully treated) that meet Criterion a (i.e., active-phase symptoms) and may include periods of prodromal or residual symptoms. During these prodromal or residual periods, the signs of the disturbance may be manifested by only negative symptoms or two or more symptoms listed in Criterion a present in an attenuated form (e.g., odd beliefs, unusual perceptual experiences).

Schizoaffective and mood disorder exclusion: Schizoaffective disorder and mood disorder with psychotic features have been ruled out because either (1) no major depressive, manic, or mixed episodes have occurred concurrently with the active-phase symptoms; or (2) if mood episodes have occurred during active-phase symptoms, their total duration has been brief relative to the duration of the active and residual periods.

Substance/general medical condition exclusion: The disturbance is not due to the direct physiological effects of a substance (e.g., a drug of abuse, a medication) or a general medical condition.

Relationship to a pervasive developmental disorder: If there is a history of autistic disorder or another pervasive developmental disorder, the additional diagnosis of schizophrenia is made only if prominent delusions or hallucinations are also present for at least a month (or less if successfully treated).

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ALL of these criteria have to be met for a diagnosis fo Schizophrenia…. typically negative symptoms are there all the time and positive symptoms come and go….

No symptom of schizophrenia is positive. If your mind is detached from reality, you're going to have a hard time in this world unless you get treatment.

Most people with schizophrenia have psychosis which means they lose contact with reality, if you have simple schizophrenia then you have mainly negative symptoms.