Is it possible for someone to work from home for a week and not go out at all or see anyone and still get the flu or is it more likely a cold?

Colds are still transmitted by viruses, similar to how flu is transmitted. Cold like symptoms may be due to non infectious conditions among which allergies are notorious.

Agree with Dr. P. If you have not had contact with any person for a week, the possibilities that your symptoms are from allergy or it is from an infection acquired before one week. Covid can have an incubation period of 10-12 days.