No, you can't use "own" that way. Also, the use of the plural "them" or "their" in reference to a singular noun, as Slipstream is recommending, is considered informal usage (it's perfectly valid in everyday speech), but it's not recommended for formal (i.e., published non-dialogue) writing.
What I usually do is change gender for each example - when I am talking about my first user, I might use "her" to suggest that the user is a woman, then when talking about a second user, use "him" to suggest that the user is a man.
EDIT: Since there is some controversy on the subject of "Singular they" here, let me cite the wikipedia article on the subject, http://en.wikipedia.org/wiki/Singular_they, and include a passage which that article quotes from the Chicago Manual of Style:
A singular antecedent requires a singular referent pronoun. Because "he" is no
longer accepted as a generic pronoun referring to a person of either sex, it
has become common in speech and in informal writing to substitute the third-person
plural pronouns "they," "them," "their," and "themselves," and the nonstandard
singular "themself." While this usage is accepted in casual contexts, it is still
considered ungrammatical in formal writing. . . . Employing an artificial form
such as "s/he" is distracting at best, and most readers find it ridiculous. There
are several better ways to avoid the problem. For example, use the traditional,
formal "he" or "she," "him" or "her," "his" or "her," "himself" or "herself."
[Added quotation marks around the example words to make it easier to read.]
The Wikipedia article also notes this apology on the Chicago Manual website: http://www.chicagomanualofstyle.org/qanda/data/faq/topics/Pronouns/faq0018.html .
Another way to approach this would be to use plural "they": to write "The users attempt to maximize their own capacity."