On this point, the Church does not make a distinction between Catholics and non-Catholics:
from the Catechism of the Catholic Church:
2283 We should not despair of the eternal salvation of persons who have taken their own lives. By ways known to him alone, God can provide the opportunity for salutary repentance. The Church prays for persons who have taken their own lives.
Let us pray for your cousin:
God, lover of souls,
you hold dear what you have made
and spare all things, for they are yours.
Look gently upon your servant, Stillsosola’s cousin,
and by the Blood of the cross
forgive his sins and failings.
Remember the faith of those who mourn
and satisfy their longing for that day
when all will be made new again
in Christ, our risen Lord,
who lives and reigns with you forever and ever.
Amen.