According to the US National Institutes of Health (NIH) (emphasis added):
Researchers used to think that eye color was determined by a single gene and followed a simple inheritance pattern in which brown eyes were dominant to blue eyes. Under this model, it was believed that parents who both had blue eyes could not have a child with brown eyes. However, later studies showed that this model was too simplistic. Although it is uncommon, parents with blue eyes can have children with brown eyes. The inheritance of eye color is more complex than originally suspected because multiple genes are involved. While a child’s eye color can often be predicted by the eye colors of his or her parents and other relatives, genetic variations sometimes produce unexpected results.
In the question, you correctly noted that "The image also says if you have brown eyes, your parents can't both have blue eyes." This is contrary to the NIH's statement.
Thus, the figure is invalidated.
Are these restrictions true?
No, the second restriction is not true. The figure is inaccurate and at least one mistake exists.