Yes, you should "check the box" although it will be obvious given your parents' employment and education level that you are a first generation college student and, based on place of birth and citizen status, an immigrant from Mexico who is now here legally.
Here's a stripped down version of your statement:
In 2001 (or whatever year is correct), my father closed his hardware store in Mexico and illegally immigrated to the United States in search for more stable income. One year later, my mother, three older brothers, and I joined him. I was nine years old and struggled to learn English. My parents only had an an elementary school education and lacked English fluency; they eventually found full employment as a car mechanic and waitress. From a young age I became familiar with the subsidized charity care at our community clinic while accompanying them for their appointments for my mother's diabetes and my father's COPD. We eventually began the process to naturalization which has been a blessing.
Most of what I cut out will be included in the Yes/No questions that accompany the disadvantaged section (e.g. work before age 18).