Big toe sore, skin red, squeezed and pus came out side of toe nail. is it a possible ingrown toe nail ? infection ?

Yes: Probably an ingrown nail due to wearing ill fitting shoes. Needs to be treated with soaks with Epsom salts and wearing properly fitting shoes that do not cause pressure on the toe box if you have diabetes sees doctoras this pilfer serious.

Dr Pack agreed

Paronychia: This is an infection along the border of the nail. As long as this drains and you keep it clean it should heal on its own. Sometimes this is due to ingrown nail and sometimes to trauma or to nail biting.

Dr Chuu agreed

Yes, and yes: When there is pus, there is infection.

Toe infection: Yes. This is an infection, and it can be from an ingrown nail. Clean the toe. Soak it in sitz bath and see your doctor soon for antibiotics. Wear looser shoes to avoid pressure on the wound.

Dr Raff and another doctor agreed