I'm Byzantine. Can I still celebrate All Saints' and All Souls'?

Yes. You guessed it. Byzantine Catholics have different days to celebrate All Saints' Day and All Souls' Day.

Eastern Christians of the Byzantine Tradition commemorate all saints collectively on the first Sunday after Pentecost, All Saints' Sunday.

The feast of All Saints achieved great prominence in the ninth century, in the reign of the Byzantine Emperor, Leo VI "the Wise" (886–911). His wife, Empress Theophano commemorated on 16 December, lived a devout life. After her death in 893, her husband built a church, intending to dedicate it to her. When he was forbidden to do so, he decided to dedicate it to "All Saints", so that if his wife were in fact one of the righteous, she would also be honored whenever the feast was celebrated. According to tradition, it was Leo who expanded the feast from a commemoration of All Martyrs to a general commemoration of All Saints, whether martyrs or not. (from wikipedia- sorry- and the clip art was drawn by artist Enid Chadwick...)

and what about All Souls' Day?

It isn't one day.

On the day before the Sunday of the Last Judgement (Meatfare Sunday before the Great Lent), and in close connection with the theme of this Sunday, there is a universal commemoration of the dead "from the ages". This is the First All Souls Saturday. There are additional special commemorations of the dead on the second, third and fourth Saturdays during the Great Fast as well as on the Saturday before Pentecost Sunday. 

Before we call to mind the Second Coming of Christ next Sunday, we commend to God all those departed before us, who are now awaiting the Last Judgement. In the liturgical celebrations for these Saturdays there is a strong sense of the continuing bond of mutual love that links together all the members of the Church, whether alive or dead. For those who believe in the risen Christ, death does not constitute an impassable barrier, since all are alive in Him; the departed are still our brothers and sisters, members of the same family with us, and so we are conscious of the need to pray insistently on their behalf.

At the local parish the commemoration of the dead takes the form of special prayers. These prayers include the reading of the names of all those who have fallen asleep in the Lord. These days are devoted to prayer for departed relatives and others among the faithful who might not be commemorated specifically as Saints. The Divine Services on these days have special hymns added to them to commemorate the departed. There is normally a Parastas (memorial service) served either after the Divine Liturgy on Saturday morning, or after Vespers on Friday evening, for which sweet bread and/or koliva (a dish made of boiled wheatberries and honey) is prepared and placed in front of the cross or icon before which the Parastas is served. After the service, the priest blesses the sweet bread and/or koliva and it is then eaten as a memorial by all present.

So, in answer to the Byzantine Catholic who posed the question- of course, you can commemorate these days on the Latin calendar! I did, but I will first remember my own traditions as a Byzantine Catholic. For the health and beauty and truth of the Church, let us not forget our traditions wherever we live.