Can botox solve a situation where a person's eyes have the "staring", "frightened", or "surprised" look?

Thank you for your question qpalzm27. I understand your concern. Botox is a purified protein used to address wrinkles associated with facial expression. When injected into the skin Botox will relax the muscles and smoothen out the overlying wrinkles. The most common areas of treatment are in the upper face. These include the horizontal lines seen on the upper forehead when one raises the brows, the vertical lines seen between the brows when one frowns (frown lines), and the crow's feet seen around the eyes when one smiles.
It is difficult to make an assessment without a formal exam in the office where I can view the skin in multiple angles and observe wrinkles formed with facial expression. However, I hope the information provided here is useful.The appearance of frown lines is variable from person to person. Usually they are in a vertical orientation but sometimes they can be oblique and sometimes a horizontal line is seen across the upper portion of the nasal bridge. The most common variation is to have two parallel lines, hence the common term "11's" to describe them. But some people have one, three, or more lines. Some people also will see a bulge in the area when contracting the muscle. This bulge is smoothened out after a Botox treatment.
The pivotal studies used 20 units of Botox in the glabella area between the brow to address the frown lines. These studies demonstrated that the peak results are seen 14 days after the treatment and most people enjoy their results for 3-4 months. At that point, a maintenance treatment is recommended. While some people start to see results as early as the next day, it can take two weeks for full results so patience is important. We have our patients return two weeks after their treatment to assess their results via photographs. At that time additional units may be added if necessary. 
When patients present with concerns about specific wrinkles in the upper face such as the frown lines, I always assess and usually treat the other wrinkles in the upper face including the horizontal forehead wrinkles and crow’s feet. The reason is to maintain a harmonious and natural balance to the appearance of the upper face. The muscles in the glabella which are responsible for the formation of the frown lines serve to pull down on the inner brows when a person frowns. Similarly, the muscles around the eyes which are responsible for the formation of the crow’s feet seen when smiling (orbicularis oculi) are also responsible for pulling down on the brows, but in this case the outer brows. On the other hand, the muscle in the upper forehead which is responsible for the formation of the horizontal forehead wrinkles when a person raises the brows (frontalis) serves to pull the brows upward. If one of these muscles is relaxed with Botox and the others are not, the other muscles will still be active. This will lead to an unbalanced and unnatural appearance of the upper face where the brow can be elevated, depressed, or uneven in shape or position. Therefore, I recommend that my patients take a global approach to the upper face when receiving Botox treatments and treat all three of these areas. Sometimes bunny lines on the bridge of the nose are also treated.
Sometimes after a Botox treatment a peaked appearance of the eyebrow is seen. This is sometimes called the Spock or Mephisto look, which makes a person appear surprised or sinister. It is also called the Jack Nicholson look, after the appearance of his eyebrows. This can occur if the muscles that pull the eyebrow down are relaxed with Botox. It can also occur if the upper forehead only is treated. In this case the inner portions of the muscles are relaxed preferentially over the outer portions of the muscle. This allows the outer portions of the muscle to pull up on the tail of the eyebrow. While some people desire this appearance, others do not. This effect can be corrected by placing a small amount (1-2 units) of Botox in the upper forehead approximately midway between the eyebrow and the hair line at the peak of the brow. This relaxes the muscle that pulls up on the tail of the eyebrow, thereby restoring a natural appearance.
With any injection, there are risks such as pain, bleeding, bruising, redness, swelling, tenderness, and infection. We take special precautions to minimize these risks such as using a painless technique. We also treat bruises with a laser as early as the next day. Bruises usually resolve within two weeks if untreated. After a laser treatment bruises usually resolve in 1-3 days, but may still take two weeks for full resolution. In addition to these risks, with Botox there is also a risk that the product spreads or diffuses to nearby structures. Spread to nearby muscles can lead to side effects such as the temporary appearance of droopy eyelids or a temporary asymmetric smile.

After a Botox treatment, I recommend that my patients avoid heat exposure, alcohol consumption, and strenuous exercise for 24 hours and not lie down flat for four hours. This is because all of these activities may possibly lead to the movement of the product and consequent side effects such as droopy eyelids. Although there have been no definitive studies shown that these actually occur and some physicians do not provide such aftercare instructions, in the absence of data I err on the side of caution and recommend that my patients avoid such activities.

Please consult with a doctor for specific recommendations. Good luck!

Botox is a wonderful product and can transform facial expression by not making you look "angry" or "tired".  The best way to see if Botox would work for you is to post a photo or consult a board certified dermatologist who has experience with Botox injections.  What you are describing can probably be improved through a combination of Botox and fillers like Restylane.