Today would be 3 days late for my period. I was sick for 4 days last week (June 1st-4th). Should I do a pregnancy test? How come it's 3 days late?

Depends. If you had unprotected sex since your last period, it would be prudent to do a pregnancy test. If it is negative, repeat the test in 4-5 days. Use first morning urine and follow the instructions for the test carefully. If you do not wish to be pregnant, use contraception all the time, every time. You may consider implanted contraceptive, or IUD. Practice safe sex. Get HPV vaccine.