Stroke: This is the most likely diagnosis due to the sudden onset of symptoms affecting both extraocular movements and facial movements, suggesting a lesion in the brainstem or a large cerebral hemisphere infarct affecting cranial nerve nuclei or their pathways.

Multiple Cranial Neuropathies: Conditions such as Guillain-Barré syndrome can lead to weakness or paralysis of extraocular and facial muscles due to inflammation of the nerves.
Myasthenia Gravis: An autoimmune disease that can cause fluctuating weakness of muscles, including those controlling eye and facial movements, due to antibodies blocking acetylcholine receptors.
Thyroid Ophthalmopathy: Although primarily associated with exophthalmos and eyelid retraction, in severe cases, it can cause restrictive strabismus affecting EOM.

Brainstem Tumor or Cyst: Lesions in the brainstem can compress cranial nerve nuclei or their pathways, leading to deficits in EOM and facial movements. Missing this diagnosis could lead to significant morbidity or mortality.
Cavernous Sinus Thrombosis: A rare but potentially life-threatening condition that can cause multiple cranial neuropathies, including those affecting EOM and facial movements, due to thrombosis in the cavernous sinus.
Wernicke's Encephalopathy: A condition caused by thiamine deficiency, often seen in alcoholics, which can present with ophthalmoplegia, among other symptoms, and requires urgent treatment to prevent permanent damage.

Mitochondrial Myopathies: A group of disorders that can affect the muscles, including those involved in eye movements, due to mitochondrial dysfunction.
Chronic Progressive External Ophthalmoplegia (CPEO): A rare mitochondrial disorder characterized by a gradual decline in eye movements.
Moebius Syndrome: A rare congenital disorder characterized by facial paralysis and impaired eye movements due to abnormal development of the cranial nerves.