Can adrenal insufucienty cause high white blood count?

No: There may be an increased # of eosinophils.

Not really: Patients with addisons disease (severe adrenal insufficiency) are often very sick. If the adrenal insufficiency comes as a result of sepsis (overwhelming infection), the WBC would be high. But most addison's is autoimmune, and the WBC is not high. Sometime one of the WBC (eosinophils) can be high. There can be other associated autoimmune problems that could affect wbc.