Yes, women can have gout.
More Info: Gout is an inflammatory arthritic disease brought on by a build-up of uric acid in the body. Uric acid is present in both men and women and is simply the by-product of the break-down of purines that are present in many foods and found naturally in the body’s tissue.
Though women can get gout, men are seven to nine times more likely to suffer from gout, which is why it is often referred to as a man’s disease. Each year gout strikes 3.4 million American men making it the most common form of inflammatory arthritis for men over forty. 
What Is Uric Acid?
Uric acid is a nitrogenous waste produced in peroxisomes within the cells of the body as a byproduct of nucleic acid metabolism. This chemical is not especially soluble in water and tends to precipitate in solution, forming sharp sodium urate crystals. These crystals contribute to the formation of kidney stones and can deposit in joints, resulting in gout. Despite its potential for harm, the kidneys reclaim uric acid to reuse. This is because the chemical is a powerful antioxidant that can protect cells from damage, and it is useful to the immune system in identifying damaged cells.
What Are Purines?
Purines are chemical compounds that are broken down into uric acid. You body is able to produce most of the purine that is needed, but there are some protein-containing foods that are high in purine. When an overindulgence of these foods has been ingested into the body, it raises the uric acid and results into a condition known as gout. Purine can be found in high doses in foods such as liver, beef kidney, sweetbread, sardines, game meats, alcohol, gravy, brains, and mackerel. Moderate amounts can be found in foods including dried beans, peas, oatmeal, poultry, mushrooms, seafood, pork, and asparagus.
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“Uric acid – blood – Overview.” University of Maryland Medical Center | Home. N.p., n.d. Web. 29 June 2011. <http://www.umm.edu/ency/article/003476.htm>.
“Introduction to DNA Structure.” Biology Learning Center at the University of Arizona. N.p., n.d. Web. 29 June 2011. <http://www.blc.arizona.edu/molecular_graphics/dna_structure/dna_tutorial.html>.