It could be a bug bite, but if it looks more like a zit then there is some type of infection going... Kinda sounds like a staph infection, I would see a doctor.
The bug/insect bite can bot be diagnosed without having a look on the affected site. Usually all insect bite have same symptoms but can differ in severity. You can apply hydrocortisone cream and take anti allergy medicines or visit to a doctor for proper diagnosis and treatment.
Chances are it is a parasite. Scabies isn't uncommon. They burrow under the skin, and the fecal matter they leave behind causes small bumps on the surface of the skin. Generally it can be cured by a cream that a doctor will prescribe to you, but it can take up to many weeks before the itchy bumps will go away.