Can a non-baptized publisher date or marry a person who is not a Jehovah's Witness?
It is worth noting that a person who has not been baptised as a Jehovah’s Witness but attends the meetings and shares in the preaching activity is known as “a publisher”. Unbaptised publishers are therefore seen as potential members. Since they have not yet been baptised, they are not officially identified as one of Jehovah’s Witnesses.
The general advice that is given in official publications is this:
Why Jehovah’s Witnesses believe they should marry only within their religion: The Witnesses heed the Bible’s instruction to marry “only in the Lord,” that is, to marry a person who shares their faith. (1 Corinthians 7:39) This command is both Biblical and practical. For example, a 2010 article in the Journal of Marriage and Family said that “couples in which partners share common religious affiliations, practices, and beliefs” tend to have higher quality relationships. Source: Do Jehovah’s Witnesses Break Up Families or Build Them Up?
Even a baptized Jehovah’s Witness CAN date a non witness. It is discouraged as seen in @lesleys answer.
But dating is not grounds for disfellowshipping so long as sexual immorality is avoided.
A baptized witness who dates a non witness would not be considered a member in good standing and would not be considered for special privileges of service.
Similarly a person who begins a relationship with a non Jehovah’s Witness while an unbaptized publisher would likely no longer qualify as one since he/she is purposefully seeking out association with a non witness and thus not meeting part of the requirements for unbaptized publisher ie:
Does he/she see the value of avoiding unwholesome association? (1 Cor. 15:33).
In such a case the congregation would continue to embrace the newly interested person and would hope that the relationship blossomed into a happy marriage with both partners becoming active JWs.
If the couple marry the newly interested person would again qualify to be an unbaptized publisher and could progress towards baptism even if the spouse does not share an interest in that endeavor.
If a person is living with a significant other and wishes to advance toward becoming an unbaptized publisher they are encouraged to marry to dignify the relationship legally and in the eyes of God. (See this related Q/A)
Of course this will require a deep conversation with SO since being married to one who has vastly different religious beliefs can be difficult but not impossible. (There are many such marriages among JWs)
All of this may take some time to sort out and until it is resolved by marriage or parting ways the newly associated one would not qualify to be an unbaptized publisher or to advance to baptism as a Jehovah’s Witness.