In "Everyday Use" the inner and outer conflicts are interrelated. They both involve the female culture of the Johnson family, as symbolized by the heirlooms of the quilts and the butter churn.
The external conflict is mainly between Mrs. Johnson and Dee over who should receive the heirlooms. Also of importance is Dee's dress and language, both of which reveal her trendy new Pan-African culture--a refusal of her native domestic, feminine, agrarian American culture. Her new afro, African garb, and Islamic greetings are signs that she has turned her back of her mother's family traditions.
All three Johnson women face internal conflicts. Mrs. Johnson battles her desire to be honored publicly as a matriarch of the family (on the Johnny Carson show). She also must protect the family traditions by rewarding Maggie, the younger daughter, and not Dee, the older. Obviously, it is tough for a mother to forsake a birthright.
Maggie feels shame for not being as pretty and educated as her older sister. Her inferiority complex comes from having been literally burned by her old house. As such, she will forever be a domestic in one. But, at the end, she feels great joy at having been given the quilts, and she is honored as the future matriarch of the family.
Dee does not exhibit much internal conflict. If she had, she might have seen how arrogant and selfish she is in her demands for the heirlooms. She wants to display them as relics of her African bloodline, not her immediate domestic culture. As such, she becomes an impostor to her mother, one who doesn't deserve the status of matriarch.